text:
\section{Introduction}

Can $\omega_2$ satisfy the tree property and also carry a saturated ideal?  This question is natural for anyone concerned with ``large cardinal properties'' that can hold of small cardinals.  It turns out to be surprisingly difficult to answer.  In \cite{coxeskew}, Cox and the author showed that if this situation is consistent, then (a) it requires the continuum to be at least $\omega_3$, and (b) it cannot be forced via a ``Kunen-style'' generic lifting of an almost-huge embedding, which is presently the only known method to produce models with saturated ideals on $\omega_2$.  This means that, if these properties of $\omega_2$ are mutually consistent, a proof will require rather novel methods.

As a step towards resolving this question, we explore the relationship between the side-conditions forcing method of Neeman \cite{neeman} and weak saturation properties.  We move the goalposts back a little and reconcile the tree property with the weak Chang's Conjecture (wCC), a common consequence of both Chang's Conjecture and the existence of a saturated ideal that has been extensively studied for example in \cite{aspero,coxlaver,dk,dl}.  Along the way, we reduce the upper bound on the consistency strength of wCC at $\omega_2$ down from almost-hugeness to $(+2)$-subcompactness.  However, we identify a barrier to extending these methods beyond wCC at $\omega_2$, implying that yet more novelty will be needed to bring saturation and the tree property together.  This barrier is an unexpected structural constraint concerning stationary sets of models connected to supercompactness.

The structure of the paper is as follows.  In \S\ref{sec:weaksat}, we discuss the connections between saturation properties of ideals and versions of Chang's Conjecture.  In \S\ref{sec:neeman}, we introduce a modified version of Neeman's forcing and lay out its important properties.  In \S\ref{sec:ee}, we discuss a large cardinal above $0^\sharp$ but below measurable that can be used with Neeman's forcing to obtain weak presaturation at $\omega_1$.  In \S\ref{sec:wccomega_2}, we obtain a new upper bound on wCC at $\omega_2$.  In \S\ref{sec:ambitious}, we introduce a strong variant of supercompactness and discuss its connection with so-called Magidor models.  Finally, in \S\ref{sec:wcctp}, we show how to combine wCC at $\omega_2$ with the tree property and derive some limitative results.

We assume familiarity with the basics of stationary sets, large cardinals, forcing, and the notions of properness and strong properness for a class of models.  We also assume a strong familiarity with \cite{neeman}, as we will rely on it.  To fix some notation, if $\kappa$ is a cardinal and $X$ is a set, we use $[X]^{<\kappa}$ to denote $\{ z \subseteq X : |z| < \kappa \}$, and if $\kappa \subseteq X$, we use $\mathcal{P}_\kappa(X)$ to denote $\{ z \subseteq X : |z| < \kappa \wedge z \cap \kappa \in \kappa \}$.  For a set of ordinals $x$, we let $\ot(x)$ denote its order-type.
  $H_\kappa$ denotes the collection of all sets of hereditary cardinality less than $\kappa$.  For a structure $\frak A$ on $X$ carrying a well-order and $Y \subseteq X$, we write $\sk^{\frak A}(Y)$ for the Skolem hull of $Y$, i.e.\ the closure of $Y$ under the definable Skolem functions.

If $\kappa$ is a regular cardinal, $T$ is a \emph{$\kappa$-tree} when it is a partially ordered set, well-ordered below any given node, such that the rank of any given node is ${<}\kappa$, and if $T_\alpha$ is the set of nodes of rank $\alpha$, then $|T_\alpha|<\kappa$.  A $\kappa$-tree is \emph{Aronszajn} if it has no cofinal branch, i.e.\ a linearly ordered subset of order-type $\kappa$.  If $\kappa = \mu^+$, a $\kappa$-tree $T$ is \emph{special} if there is a function $f : T \to \mu$ such that if $a <_T b$, then $f(a) \not= f(b)$.  Clearly, special trees are Aronszajn.

\section{Weak saturation properties}
\label{sec:weaksat}
Foreman \cite{foreman} proved an equivalence between a normal fine ideal being saturated and a statement about elementary submodels.  Inspired by this, we identify some weakenings of saturation that also have equivalences in terms of elementary submodels, in the hopes that a forcing such as Neeman's, which is built from sequences of models, may be able to achieve desired results by directly manipulating properties of collections of elementary submodels.  Let us first recall some basic notions.

$\ensuremath{\mathcal{I}}$ is an \emph{ideal} over a set $Z$ when $\ensuremath{\mathcal{I}} \subseteq \mathcal{P}(Z)$, and $\ensuremath{\mathcal{I}}$ is closed under subsets and pairwise unions.  For a cardinal $\kappa$, $\ensuremath{\mathcal{I}}$ is \emph{$\kappa$-complete} when it is closed under unions of size ${<}\kappa$.  We say a set $A \subseteq Z$ is \emph{$\ensuremath{\mathcal{I}}$-positive} when it is not in $\ensuremath{\mathcal{I}}$.  We denote the collection of all such sets as $\ensuremath{\mathcal{I}}^+$, and we let $\ensuremath{\mathcal{I}}^d$ denote the filter dual to $\ensuremath{\mathcal{I}}$.  If $Z \subseteq \mathcal{P}(X)$, we say that $\ensuremath{\mathcal{I}}$ is \emph{fine} when $\{ z \in Z : x \in z \} \in \ensuremath{\mathcal{I}}^d$ for all $x \in X$, and \emph{normal} when for every $\ensuremath{\mathcal{I}}$-positive $A$ and every function $f : A \to X$ such that $f(z) \in z$ for all $z \in A$, there is $x \in X$ such that $f^{-1}[\{x\}]$ is $\ensuremath{\mathcal{I}}$-positive.  A normal fine ideal $\ensuremath{\mathcal{I}}$ on $Z \subseteq \mathcal{P}(X)$ is said to be \emph{saturated} when for every collection $\mathcal{A}$ of $\ensuremath{\mathcal{I}}$-positive sets that have pairwise intersection in $\ensuremath{\mathcal{I}}$, $|\mathcal{A}| \leq |X|.$  In other words, the Boolean algebra $\mathcal{P}(Z)/\ensuremath{\mathcal{I}}$ has the $|X|^+$-chain condition.

A set $Z \subseteq \mathcal{P}(X)$ is \emph{stationary} when for all $F : X^{<\omega} \to X$, there is $z \in Z$ closed under $F$.  The set of all $z \subseteq \mathcal{P}(X)$ closed under such an $F$ is called a \emph{closed-unbounded set} or a \emph{club}.  It is well-known that for all stationary sets $Z$, the collection of all nonstationary subsets of $Z$ forms the smallest normal fine ideal over $Z$.  The ideal of nonstationary subsets of $Z$ will be denoted by $\ns_Z$, or just $\ns$ when $Z$ is clear from context.  If $Z$ is a regular cardinal $\kappa$, this notion of club coincides with the usual order-topological notion.

Suppose $\pi : Z_1 \to Z_0$ and $\ensuremath{\mathcal{J}}$ is an ideal on $Z_1$.  The map $\pi$ induces an ideal $\ensuremath{\mathcal{I}}$ on $Z_0$, where we put $A \in \ensuremath{\mathcal{I}}$ iff $\pi^{-1}[A] \in \ensuremath{\mathcal{J}}$.  If $\ensuremath{\mathcal{J}}$ is normal, fine, and $\kappa$-complete, then so is $\ensuremath{\mathcal{I}}$.  If $Z_i \subseteq \mathcal{P}(X_i)$, and $\pi$ is the map $z \mapsto z \cap X_0$, then we say that $\ensuremath{\mathcal{I}}$ is the \emph{canonical projection} of $\ensuremath{\mathcal{J}}$.
For a normal fine ideal $\ensuremath{\mathcal{I}}$ on $Z \subseteq \mathcal{P}(X)$ and $\theta>2^{|Z|}$, Foreman \cite{foreman} defined a model $M \prec H_\theta$ to be \emph{$\ensuremath{\mathcal{I}}$-good} when $\ensuremath{\mathcal{I}} \in M$ and $M \cap X \in A$ for every $A \in \ensuremath{\mathcal{I}}^d \cap M$.  He showed:

\begin{theorem}[Foreman]
Suppose $\ensuremath{\mathcal{I}}$ is a normal ideal on a regular cardinal $\kappa$ and $\theta>2^\kappa$ is regular.
\begin{enumerate}
\item If $\ensuremath{\mathcal{G}}$ is the set of $\ensuremath{\mathcal{I}}$-good $M \prec H_\theta$, then $\ensuremath{\mathcal{G}}$ is stationary, and $\ensuremath{\mathcal{I}}$ is the canonical projection of $\ns \restriction \ensuremath{\mathcal{G}}$.
\item $\ensuremath{\mathcal{I}}$ is saturated if and only if for every $\ensuremath{\mathcal{I}}$-good $M \prec H_\theta$ and every maximal antichain $\mathcal{A} \subseteq \mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}$ in $M$, there is $a \in \mathcal{A} \cap M$ such that $M \cap \kappa \in a$.
\end{enumerate}
\end{theorem}

It is well-known that saturated ideals produce well-founded generic ultrapowers, which are moreover highly closed from the perspective of the generic extension.  But these properties can be obtained from a weaker notion.
A normal ideal $\ensuremath{\mathcal{I}}$ on a regular cardinal $\kappa$ is called \emph{presaturated} when for all $\mu<\kappa$, all $\ensuremath{\mathcal{I}}$-positive $B$, and all collections of antichains $\langle \mathcal{A}_i : i < \mu \rangle$, there is an $\ensuremath{\mathcal{I}}$-positive $C \subseteq B$ such that $|\{ a \in \mathcal{A}_i : a \cap C \in \ensuremath{\mathcal{I}}^+ \}| \leq \kappa$ for all $i<\mu$.
It is not hard to show that forcing with a presaturated ideal on $\kappa$ preserves $\kappa^+$.  Thus if $\kappa$ is a successor cardinal, any  generic embedding arising from such an ideal must send $\kappa$ to $(\kappa^+)^V$.

Woodin \cite{woodin} calls a normal ideal $\ensuremath{\mathcal{I}}$ on a successor cardinal $\kappa$ \emph{weakly presaturated} just when the above property occurs, i.e.\ $\Vdash_{\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}} j_{\dot G}(\kappa) = (\kappa^+)^V$.  This can be explicated in terms of \emph{canonical functions}.
Recall that the canonical functions on a regular cardinal $\kappa$ represent ordinals $\alpha<\kappa^+$ in any generic ultrapower coming from a normal ideal on $\kappa$, and they can be defined (up to equivalence mod clubs) as follows:  For $\alpha<\kappa^+$, pick any surjection $\sigma_\alpha : \kappa \to \alpha$, and define the $\alpha^{th}$ canonical function $\psi_\alpha$ by $\psi_\alpha(\beta) = \ot(\sigma_\alpha[\beta])$ for all $\beta < \kappa$.

\begin{lemma}
\label{wps-equivs}
Suppose $\ensuremath{\mathcal{I}}$ is a normal ideal on a successor cardinal $\kappa$.  The following are equivalent:
\begin{enumerate}
\item\label{wps-genult} $\ensuremath{\mathcal{I}}$ is weakly presaturated.
\item\label{wps-dense} For all $\ensuremath{\mathcal{I}}$-positive $A \subseteq \kappa$ and all $f : \kappa \to \kappa$, there is $\alpha<\kappa^+$ and an $\ensuremath{\mathcal{I}}$-positive $B \subseteq A$ such that $f(\beta) < \psi_\alpha(\beta)$ for all $\beta \in B$.
\item\label{otdom} There is $\theta > \kappa$ and a stationary set $\mathcal{S} \subseteq \{ M \prec H_\theta : M \cap \kappa \in \kappa \}$ such that:
\begin{enumerate}
\item $\ensuremath{\mathcal{I}}$ is the canonical projection of $\ns \restriction \mathcal{S}$.
\item For all $A \in \ensuremath{\mathcal{I}}^+$ and all $f : \kappa \to \kappa$, the set
$$\{ M \in \mathcal{S} : M \cap \kappa \in A \wedge f(M \cap \kappa) < \ot(M \cap \kappa^+)  \}$$
is stationary.
\end{enumerate}
\end{enumerate}
\end{lemma}

\begin{proof}
Suppose (\ref{wps-genult}).  Let $A$ be $\ensuremath{\mathcal{I}}$-positive and $f : \kappa \to \kappa$.  $f$ represents some ordinal $<j(\kappa)$, and thus there is some $\ensuremath{\mathcal{I}}$-positive $B \subseteq A$ deciding that for some canonical function $\psi_\alpha$, $[f]_{\dot G} < [\psi_\alpha]_{\dot G}$.  This implies that for $\ensuremath{\mathcal{I}}$-almost-all $\beta \in B$, $f(\beta) <\psi_\alpha(\beta)$, establishing (\ref{wps-dense}).  On the other hand, if (\ref{wps-dense}) holds, we have that every ordinal of the generic ultrapower $<j(\kappa)$ is represented by a function $f : \kappa \to \kappa$, and the statement says that for such $f$, there is a dense set of conditions in the forcing $\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}$ deciding that $f$ represents an ordinal below $(\kappa^+)^V$, showing (\ref{wps-genult}).

Suppose (\ref{wps-dense}).  By Foreman's theorem, $\ensuremath{\mathcal{I}}$ is the canonical projection of $\ns \restriction \ensuremath{\mathcal{G}}$, where $\ensuremath{\mathcal{G}}$ is the set of $\ensuremath{\mathcal{I}}$-good $M \prec H_\theta$.  Suppose $f : \kappa \to \kappa$ and $A$ is $\ensuremath{\mathcal{I}}$-positive.  By hypothesis, there is an $\ensuremath{\mathcal{I}}$-positive $B \subseteq A$ and an $\alpha<\kappa^+$ such that $\psi_\alpha$ dominates $f$ on $B$.  There are stationary-many $M \in \ensuremath{\mathcal{G}}$ such that $M \cap \kappa \in B$.
Let $\sigma_\alpha : \kappa \to \alpha$ be a surjection defining $\psi_\alpha$.
If $M$ is any such model with $\sigma_\alpha \in M$, then
$$\psi_\alpha(M \cap \kappa) = \ot(\sigma_\alpha[M \cap \kappa]) = \ot(M \cap \alpha) < \ot(M \cap \kappa^+).$$
Thus there are stationary-many $M \in \ensuremath{\mathcal{G}}$ with $M \cap \kappa \in A$ and $f(M \cap \kappa) < \psi_\alpha(M \cap \kappa) < \ot(M \cap \kappa^+)$, establishing (\ref{otdom}).

Suppose (\ref{otdom}).  Let $A$ be $\ensuremath{\mathcal{I}}$-positive and $f : \kappa \to \kappa$.  There are stationary-many $M \in \mathcal{S}$ such that $M\cap \kappa \in A$ and $f(M \cap \kappa) < \ot(M \cap \kappa^+)$.  By Fodor's Lemma, there is an $\alpha<\kappa^+$ and a stationary $\mathcal{T} \subseteq \mathcal{S}$ such that for all $M \in \mathcal{T}$, $\alpha \in M$, $M \cap \kappa \in A$, and $f(M \cap \kappa) < \ot(M \cap \alpha)$.  We may assume that there is a surjection $\sigma_\alpha : \kappa \to \alpha$ that belongs to each $M \in \mathcal{T}$.  For all $M \in \mathcal{T}$, $f(M \cap \kappa) < \ot(M \cap \alpha) = \ot(\sigma_\alpha[M \cap \kappa]) = \psi_\alpha(M \cap \kappa)$.  Let $B = \{ M \cap \kappa : M \in \mathcal{T} \}$.  Then $B$ is an $\ensuremath{\mathcal{I}}$-positive subset of $A$, and $\psi_\alpha$ dominates $f$ on $B$.  This shows (\ref{wps-dense}).
\end{proof}

Chang's Conjecture $(\kappa_1,\kappa_0) \twoheadrightarrow (\mu_1,\mu_0)$ states that for every function $F : \kappa_1^{<\omega} \to \kappa_1$, there is a $z \subseteq \kappa_1$ closed under $F$ such that $|z| = \mu_1$ and $|z \cap \kappa_0| = \mu_0$.  If $\kappa = \mu^+$, then we abbreviate $(\kappa^+,\kappa) \twoheadrightarrow (\mu^+,\mu)$ by $\cc(\kappa)$, which is equivalent to the statement that for all $F : (\kappa^+)^{<\omega} \to \kappa^+$, there is $z \subseteq \kappa^+$ such that $z \cap \kappa \in \kappa$ and $\ot(z) = \kappa$ (see \cite{foreman}).
 According to \cite{dl}, Magidor showed that if $\kappa = \mu^+$ and $\cc(\kappa)$ holds, then there is a weakly presaturated ideal on $\kappa$. (Note that the dual of a weakly presaturated ideal is called a ``flat filter'' there.)  In fact, such an ideal is obtained as the canonical projection of the nonstationary ideal on $\mathcal{P}(\kappa^+)$ restricted to the set $\{ z : z \cap \kappa \in \kappa \wedge \ot(z) = \kappa \}$.  A result of Foreman and Magidor \cite{fm} shows that all but nonstationary-many $z$ in this set satisfy $\cf(z \cap \kappa) = \cf(\mu)$.

The \emph{weak Chang's Conjecture} at a regular cardinal $\kappa$ says:  For all functions $F : (\kappa^+)^{<\omega} \to \kappa^+$, there is $\alpha<\kappa$ such that for all $\beta<\kappa$, there is $z \subseteq \kappa^+$ closed under $F$ such that $z \cap \kappa = \alpha$ and $\ot(z) >\beta$.
  To further specify this kind of property, let us write $\wcc(\kappa,A)$ for $A \subseteq \kappa$ to assert that such $\alpha$ can be found in $A$.
  When $\kappa = \mu^+$, the result of Foreman and Magidor mentioned above tells us that $\cc(\kappa)$ implies $\wcc(\kappa,\cof(\mu))$.
  The next result shows that $\wcc(\kappa,A)$ follows from the existence of a weakly presaturated normal ideal $\ensuremath{\mathcal{I}}$ on $\kappa$ such that $A \in \ensuremath{\mathcal{I}}^d$.

\begin{lemma}
Suppose $\kappa$ is a regular cardinal and $A \subseteq \kappa$.  The following are equivalent.
\begin{enumerate}
\item\label{wcc} $\wcc(\kappa,A)$.
\item\label{wccmod} For all $f : \kappa \to \kappa$, there are stationary-many $M \in \mathcal{P}_\kappa(H_{\kappa^+})$ such that $M \cap \kappa \in A$ and $f(M\cap\kappa) < \ot(M\cap\kappa^+)$.
\item\label{canfunnodom} For all $f : \kappa \to \kappa$, there is $\alpha<\kappa^+$ such that $\{ \beta \in A : f(\beta) < \psi_\alpha(\beta) \}$ is stationary.
\end{enumerate}
\end{lemma}

\begin{proof}
Suppose $\wcc(\kappa,A)$ holds and $f : \kappa \to \kappa$.  Let $\frak A$ be any structure on $H_{\kappa^+}$ in a countable language with a well-order.
If $F : (\kappa^+)^{<\omega}\to\kappa^+$ is the result of restricting inputs and outputs of Skolem functions for $\frak A$ to ordinals, then every $z$ closed under $F$ has $\sk^{\frak A}(z) \cap\kappa^+ = z$.  By $\wcc(\kappa,A)$, there is $M \prec \frak A$ such that $M \cap
\kappa \in A$ and $f(M \cap \kappa)<\ot(M \cap \kappa^+)$.  Thus (\ref{wcc}) implies (\ref{wccmod}).

Suppose towards a contradiction that  (\ref{wccmod}) holds and (\ref{canfunnodom}) fails.  Let $\lhd$ be a well-order of $H_{\kappa^+}$, and assume that each canonical function $\psi_\alpha$ on $\kappa$ is defined using the $\lhd$-least surjection $\sigma : \kappa \to \alpha$.  Let $f : \kappa \to \kappa$ be such that for all $\alpha<\kappa^+$, $\psi_\alpha(\beta) < f(\beta)$ for all but nonstationary-many $\beta \in A$.  Let $\frak A = \langle H_{\kappa^+},\in,\lhd,f,A \rangle$.  Let $M \prec \frak A$ be such that $M \cap \kappa \in A$ and $f(M \cap \kappa)<\ot(M \cap \kappa^+)$.  Let $\alpha \in M$ be such that $f(M \cap \kappa)<\ot(M \cap \alpha)$.  Let $\sigma : \kappa \to \alpha$ be the $\lhd$-least surjection, so $\sigma \in M$ and $\psi_\alpha(\beta) = \ot(\sigma[\beta])$ for all $\beta<\kappa$. By elementarity, there is a club $C \subseteq \kappa$ in $M$ such that $\psi_\alpha(\beta) < f(\beta)$ for all $\beta \in C\cap A$.  But clearly $M \cap \kappa \in C \cap A$ and $\psi_\alpha(M \cap \kappa) = \ot(M \cap \alpha)>f(M \cap \kappa)$, a contradiction.

Suppose towards a contradiction that (\ref{canfunnodom}) holds but $\wcc(\kappa,A)$ fails.  Let $F : (\kappa^+)^{<\omega} \to \kappa^+$ be such that for all $\alpha \in A$, the set $\{ \ot(z) : z \cap \kappa = \alpha \wedge F[z^{<\omega}] \subseteq z \}$ is bounded below $\kappa$.  Let $f : A \to \kappa$ return such a bound.  Let $\frak A$ be a structure on $H_{\kappa^+}$ in a countable language such that if $M \prec \frak A$, then $M \cap \kappa^+$ is closed under $F$.  By (\ref{canfunnodom}), let $\alpha<\kappa^+$ be such that $B = \{ \beta \in A : f(\beta) < \psi_\alpha(\beta) \}$ is stationary.  Let $M \prec \frak A$ be such that $M \cap \kappa \in B$ and $\alpha,f \in M$.  Then $f(M \cap \kappa)<\psi_\alpha(M \cap \kappa) = \ot(M \cap \alpha)$, contrary to the definition of $f$.
\end{proof}

\begin{corollary}
\label{wps-wcc}
Suppose $\kappa$ is a regular cardinal and $A \subseteq \kappa$ is stationary.  The following are equivalent.
\begin{enumerate}
\item $\ns_\kappa \restriction A$ is weakly presaturated.
\item $\wcc(\kappa,B)$ holds for every stationary $B \subseteq A$.
\end{enumerate}
\end{corollary}

\section{Modified Neeman forcing}
\label{sec:neeman}

Let us recall the definition of Neeman's model sequence poset \cite{neeman}.  We fix some transitive set $K$ satisfying a sufficient amount of ZFC.  In our applications, $K$ will always be $H_\theta$ for some regular $\theta$.  We fix two classes of elementary submodels of $K$, the small models $\mathcal{S}$ and the transitive models $\mathcal{T}$, and a cardinal $\kappa$ with the following properties:
\begin{itemize}
\item $\mathcal{S} \cup \mathcal{T} \subseteq K$.
\item $\kappa+1 \subseteq M$ for all $M \in \mathcal{S} \cup \mathcal{T}$.
\item If $M,N \in \mathcal{S}$ and $M \in N$, then $M \subseteq N$.
\item If $W \in \mathcal{T}$, $M \in \mathcal{S}$, and $W \in M$, then $M \cap W \in \mathcal{S} \cap W$.
\item Each $W \in \mathcal{T}$ is transitive and ${<}\kappa$-closed.
\end{itemize}
A pair $\langle \mathcal{S},\mathcal{T} \rangle$ satisfying these conditions is called \emph{appropriate} for $\kappa$ and $K$.
Usually these conditions are implied by defining $\mathcal{S}$ and $\mathcal{T}$ such that for some regular cardinal $\lambda\geq\kappa$, every $M \in \mathcal{S}$ has $M \cap \lambda \in \lambda$ and $|M|<\lambda$, while every $W \in \mathcal{T}$ is ${<}\lambda$-closed.
The poset $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}$ consists of sets of models $s \in [\mathcal{S} \cup \mathcal{T}]^{<\kappa}$ such that the rank function is injective on $s$, and if $ \langle M_i : i < \alpha \rangle$ enumerates $s$ by order of rank, then for each $\beta,\gamma<\alpha$:
\begin{itemize}
\item $\{ i < \beta : M_i \in M_\beta \}$ is cofinal in $\beta$.
\item $s \cap M_\beta \in M_\beta$.
\item $M_\beta \cap M_\gamma = M_i$ for some $i<\alpha$.
\end{itemize}
A condition $t$ is stronger than a condition $s$ when $t \supseteq s$.
Neeman also introduced a \emph{decorated} version of this poset, which enforces some continuity of the generic object added.  We will slightly modify these decorations as follows.  We define $\mathbb P^{\mathrm{dec}*}_{\kappa,\mathcal{S},\mathcal{T}}$ to consist of pairs $\langle s,f \rangle$ where:
\begin{enumerate}
\item $s \in \mathbb P_{\kappa,\mathcal{S},\mathcal{T}}$.
\item\label{dec*} $f$ is a function with $\dom(f) \in [s \times \kappa]^{<\kappa}$.  If $\langle M,\alpha \rangle \in \dom(f)$, then $f(M,\alpha) \in M^* \cap W$, where $M^*$ is the successor of $M$ in $s$ if it exists, $M^* = K$ if $M$ is the largest model in $s$, and $W$ is the smallest member of $\mathcal{T}$ such that $M \in W$.
\item\label{decrest} $f \in K$, and for each $M \in s$, $f \restriction M \in M$.
\end{enumerate}
In order for (\ref{dec*}) to make sense, we assume that $\mathcal{T}$ is linearly ordered by $\in$ and $\subseteq$-cofinal in $K$.
We note that by Claim 2.34 of \cite{neeman}, if $\langle s,f \rangle$ is a condition and $M \in s$, then $\ran(f \restriction M) \subseteq M$, so (\ref{decrest}) is equivalent to saying that $f \cap M \in M$ for all $M \in s$.
We put $\langle t,g \rangle \leq \langle s,f \rangle$ when $s \subseteq t$ and $f \subseteq g$.
For a model $M \in \mathcal{S}\cup\mathcal{T}$ and a condition $p = \langle s,f\rangle$, we sometimes write $p \restriction M$ for $\langle s \cap M, f \cap M \rangle$.

Essentially, the only differences between our decorated poset and that of \cite{neeman} are that we consider a smaller class of decorations, namely those ${<}\kappa$-sized sets that are partial functions on $\kappa$, and we don't allow the decorations attached to a model $M$ to jump past the next transitive model above $M$.  One can check that the proof of strong properness for $\mathcal{S} \cup \mathcal{T}$, in particular Claim 2.38 of \cite{neeman}, still holds for this modified poset.  (Almost no change to the argument is needed to accommodate our modified decorations.)  Let us state this result:
\begin{lemma}
\label{spdec}
Suppose $\langle s,f \rangle \in \mathbb P^{\mathrm{dec}*}_{\kappa,\mathcal{S},\mathcal{T}}$ and $M \in s$.  Then $\langle s \cap M, f \cap M \rangle$ is a condition, and if $\langle t,g \rangle \in M$ and $\langle t,g \rangle \leq \langle s\cap M, f \cap M \rangle$, then $\langle t,g \rangle$ is compatible with $\langle s,f \rangle$.  Furthermore, these conditions have a greatest lower bound $\langle r,h \rangle$, where $r$ is the closure of $s \cup t$ under intersections, $r \cap M = t$, and $h = f \cup g$.
\end{lemma}

When we take a generic filter $G$ for $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}$ or $\mathbb P^{\mathrm{dec}*}_{\kappa,\mathcal{S},\mathcal{T}}$, we will say that a model $M$ ``appears in $G$''  to mean, if we are using the undecorated poset, that there is some $s \in G$ with $M \in s$, and if we are using the decorated poset, that there is some $\langle s,f \rangle \in G$ with $M \in s$.

As noted in \cite{neeman}, if each $M \in \mathcal{S}$ is ${<}\kappa$-closed then the whole forcing is ${<}\kappa$-closed.
The key points of these modifications are as follows:
\begin{enumerate}
\item The union of the decorations appearing in a generic that are attached to a model $M$ will be a surjection from $\kappa$ to the successor model $M^*$, which in typical situations must be a small model.
\item If $K = H_\theta$, $\theta^{<\theta} = \theta$, and $\mathcal{T}$ is sufficiently rich, then the poset satisfies the $\theta$-c.c.  This does not hold for the version of the decorated poset appearing in \cite{neeman}, since, for $\kappa = \omega$, it adds a club subset of $\theta$ that contains no infinite ground model set.
\item
In typical situations, the sequence of small models appearing between two consecutive transitive models will be continuous at limits of cofinality $\kappa$.
\end{enumerate}

Suppose $K = H_\theta$ for some regular $\theta$.  Then $\theta^{<\theta} = \theta$ if and only if there is a continuous $\in$-increasing sequence of transitive sets $\vec W = \langle W_\alpha : \alpha < \theta \rangle$ such that each $W_\alpha \in H_\theta$ and $\bigcup_{\alpha<\theta} W_\alpha = H_\theta$.  Let us call such a sequence a \emph{filtration} of $H_\theta$.
In case $\theta$ is an inaccessible cardinal, then we can simply take $W_\alpha=V_\alpha$.  Otherwise, it will be useful to take $\vec W$ as a predicate and consider collections of elementary submodels of $\langle H_\theta,\in,\vec W \rangle$.  When $\alpha^{<\kappa} <\theta$ for each $\alpha<\theta$, each $W_\alpha \prec \langle H_\theta,\in,\vec W \rangle$ with $\cf(\alpha) \geq \kappa$ is ${<}\kappa$-closed.  This is because for any $x \in [W_\alpha]^{<\kappa}$, there is some $\beta<\alpha$ such that $x \subseteq W_\beta$, and $[W_\beta]^{<\kappa} \subseteq W_\alpha$ by elementarity.  Thus the set of such $W_\alpha$'s can be part of an appropriate pair for $\kappa$ and $H_\theta$.

\begin{lemma}
\label{rankmeet}
Suppose $\theta$ is regular, $\vec W = \langle W_\alpha : \alpha < \theta \rangle$ is a filtration of $H_\theta$, $\frak A = \langle H_\theta,\in,\vec W \rangle$, $M \prec \frak A$, and $\alpha \in M$.  Then $M \cap W_\alpha \prec \frak A$ if and only if $W_\alpha \prec \frak A$.  Furthermore, if $W_\beta \prec \frak A$ and $\alpha = \min(M \setminus \beta)$, then $W_\alpha \prec \frak A$.
\end{lemma}

\begin{proof}
We use the Tarksi-Vaught criterion.  Suppose $W_\alpha \prec \frak A$.  Let $a \in M \cap W_\alpha$ and suppose $\frak A \models \exists x \varphi(a,x)$.  Then $W_\alpha \models  \exists x \varphi(a,x)$, and $M \models (\exists x \in W_\alpha) \varphi(a,x)$.  Thus there is $b \in M \cap W_\alpha$ such that $\frak A \models \varphi(a,b)$.

Now suppose $M \cap W_\alpha \prec \frak A$.  If $W_\alpha \nprec \frak A$, then there is some $a \in W_\alpha$ and a formula $\varphi(x,y)$ such that $\frak A \models \exists x \varphi(x,a)$, but a witness cannot be found in $W_\alpha$.  Then
$$M \models (\exists y \in W_\alpha)(\exists x \varphi(x,y) \wedge (\forall z \in W_\alpha)\neg\varphi(z,y))$$
Let $b \in M \cap W_\alpha$ witness the outermost quantifier.  By our suppositions, there is $c \in M \cap W_\alpha$ such that $\frak A \models \varphi(c,b)$.  This contradicts that $\frak A \models (\forall z \in W_\alpha)\neg\varphi(z,b))$.

Now suppose $W_\beta \prec \frak A$ and $\alpha = \min(M \setminus \beta)$.  If $W_\alpha \nprec \frak A$, there is $a \in W_\alpha$ and a formula $\varphi(x,y)$ such that $\frak A \models \exists y\varphi(a,y) \wedge  (\forall y \in W_\alpha) \neg \varphi(a,y)$.  By elementarity, there is $b \in M \cap W_\alpha = M \cap W_\beta$ such that
$M \models \exists y \varphi(b,y) \wedge (\forall y \in W_\alpha) \neg \varphi(b,y)$.
Since $W_\beta \prec \frak A$, there is $c \in W_\beta \subseteq W_\alpha$ such that $\frak A \models \varphi(b,c)$.
But this contradicts that $\frak A \models (\forall y \in W_\alpha)\neg\varphi(b,y))$.
\end{proof}

Let us say that a tuple $\langle \kappa,\lambda,\theta,\vec W,\mathcal{S},\mathcal{T} \rangle$ is \emph{nice} when:
\begin{enumerate}
\item $\kappa<\lambda<\theta$ are regular cardinals;
\item $\alpha^{<\kappa} < \theta$ for each $\alpha<\theta$;
\item $\vec W = \langle W_\alpha : \alpha <\theta \rangle$ is a filtration of $H_\theta$;
\item $\mathcal{S}$ is a set of $M \prec \frak A = \langle H_\theta,\in,\vec W \rangle$ such that each $M \in \mathcal{S}$ satisfies $|M| <\lambda$ and $M \cap \lambda \in \lambda$;
\item $\mathcal{T} = \{ W_\alpha \prec \frak A : \cf(\alpha) \geq \lambda \}$;
\item $\langle \mathcal{S},\mathcal{T} \rangle$ is appropriate for $\kappa$ and $\frak A$.
\end{enumerate}
The following is an extension of Claim 5.7 of \cite{neeman}:

\begin{lemma}
\label{allT}
Suppose $\langle \kappa,\lambda,\theta,\vec W,\mathcal{S},\mathcal{T} \rangle$ is nice.   Let $\mathbb P$ be either $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}$ or $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}^{\mathrm{dec}*}$.  Then for any $p \in \mathbb P$ and $W \in \mathcal{T}$, $p$ can be extended to include the model $W$.
\end{lemma}

\begin{proof}
We give the proof for the decorated poset; the other case is just slightly simpler.
Let $\langle s,f \rangle \in \mathbb P$ and let $W_\alpha \in \mathcal{T}$.  First suppose $s \subseteq W_\alpha$.  Since for each $\langle M,\alpha\rangle \in \dom(f)$, $f(M,\alpha)$ is required to be in the smallest $W \in \mathcal{T}$ such that $M \in W$, $f \subseteq W_\alpha$.  Since $W_\alpha$ is ${<}\kappa$-closed, $\langle s,f \rangle \in W_\alpha$.  Thus $\langle s \cup \{W_\alpha\}, f \rangle$ is a condition witnessing the claim.  If the claim fails for $W_\alpha$, then it must be witnessed by $\langle s,f \rangle$ with $s \nsubseteq W_\alpha$.
Let us assume that $\langle s,f \rangle$ is a witness to failure with $\min \{ \rank(M) : M \in s \setminus W_\alpha \}$ as small as possible.
If $M$ is of minimal rank $\geq\alpha$ in $s$, then $\rank(M) >\alpha$.  If $\alpha \in M$, then
$\langle s \cap M \cup \{W_\alpha \}, f \cap M \rangle$
is a condition in $M$ below $\langle s \cap M, f \cap M \rangle$, since $f \cap M \subseteq W_\alpha$.  Thus by Lemma \ref{spdec}, it is compatible with $\langle s,f \rangle$.

Therefore, we must have $\alpha \notin M$.  Thus $M$ must be in $\mathcal{S}$.  Let $\beta = \min(M \cap \theta \setminus \alpha)$.  Since $M$ is not cofinal in $\beta$, $\cf(\beta) \geq \lambda$.
By Lemma \ref{rankmeet}, $W_\beta \in \mathcal{T}$.  Thus $\langle s \cap M, f \cap M \rangle$ can be extended in $M$ to include $W_\beta$.
By Lemma \ref{spdec}, there is a condition $p \leq \langle s,f \rangle$ that includes $W_\beta$.  By the minimality assumption on $\langle s,f \rangle$, there is $q \leq p$ that includes $W_\alpha$.  This contradicts that $\langle s,f \rangle$ witnesses the failure of the claim.
\end{proof}

\begin{corollary}
\label{thetacc}
Under the hypotheses of Lemma \ref{allT}, $\mathbb P$ is $\theta$-c.c.
\end{corollary}

\begin{proof}
Suppose that $\mathcal{A} \subseteq \mathbb P$ is a maximal antichain.  Let $W \in \mathcal{T}$ be such that $\mathcal{A} \cap W$ is maximal in $\mathbb P \cap W$.  Let $p \in \mathbb P$ be arbitrary, and extend it to $q$ that includes $W$.  There is $a \in \mathcal{A} \cap W$ that is compatible with $q \restriction W$.   By Lemma \ref{spdec}, $a$ is compatible with $q$.  Thus $\mathcal{A} \subseteq W$.
\end{proof}

\begin{lemma}
\label{contchains}
Suppose $\langle \kappa,\lambda,\theta,\vec W,\mathcal{S},\mathcal{T}  \rangle$ is nice and $\mathcal{S}$ is stationary.
Let $G \subseteq \mathbb P^{\mathrm{dec}*}_{\kappa,\mathcal{S},\mathcal{T}}$ be generic, let $W_0 \in W_1$ be two consecutive transitive models of $\mathcal{T}$, and let $(W_0,W_1)_G$ be the set of models appearing in $G$ between $W_0$ and $W_1$.  Then $(W_0,W_1)_G$ is a subset of $\mathcal{S}$ that is:
\begin{enumerate}
\item linearly ordered by both $\in$ and $\subseteq$,
\item $\subseteq$-cofinal in $W_1$, and
\item continuous at limit points of $V$-cofinality at least $\kappa$.
\end{enumerate}
\end{lemma}

\begin{proof}
The linearity of $(W_0,W_1)_G$ follows from \cite[Claim 2.10]{neeman}.  To show that it is cofinal in $W_1$, let $\langle s,f \rangle$ be a condition such that $W_0,W_1 \in s$ and let $x \in W_1$.  By the stationarity of $\mathcal{S}$, there is $M \in \mathcal{S}$ such that $\{ \langle s,f \rangle,x \} \in M$.  Applying Lemma \ref{spdec}, there is $\langle t,g\rangle \leq \langle s,f \rangle$ with $M \in t$.  We must have $x \in M \cap W_1 \in t$.  Thus genercity implies that the union of $(W_0,W_1)_G$ is $W_1$.

Now suppose that $\langle s,f \rangle$ forces
that $M$ is a model in the interval $(W_0,W_1)_G$ whose index in the increasing enumeration following its $\in$-ordering is a limit ordinal $\eta$ such that $\cf^V(\eta) \geq \kappa$.  Let $x \in M$ be arbitrary.
Since there are no transitive models between $W_0$ and $M$, $s \cap M$ includes the interval $[W_0,M)_s$.  Since $|s| <\kappa$, there is $\langle t,g \rangle \leq \langle s,f \rangle$ such that $[W_0,M)_s$ is contained in a strict initial segment of $[W_0,M)_t$.  Let $N$ be the first model of $t$ above $[W_0,M)_s$.  Let $r \in M$ be the initial segment of $t \cap M$ up to and including $N$.  Then $r \in \mathbb P_{\kappa,\mathcal{S},\mathcal{T}} \cap M$.
Let $h$ be the function $g \restriction (r \times \kappa) \cup \{ \langle \langle N,0 \rangle, x \rangle \}$.
Then $\langle r,h \rangle$ is a condition in $M$ below $\langle s \cap M,f \cap M \rangle$.  By Lemma \ref{spdec}, it is compatible with $\langle s,f \rangle$.  For any condition $\langle s',f' \rangle$ below both $\langle r,h\rangle$ and $\langle s,f \rangle$ in which $N$ is not the largest model below $M$, $x$ is a member of the successor of $N$ in $s'$.
Since $\langle s,f \rangle$ was an arbitrary condition forcing $M$ to be at place $\eta$ in $(W_0,W_1)_G$, it follows that models appearing at such places must be the union of the models in $(W_0,W_1)_G$ appearing below them.
\end{proof}

\begin{lemma}
\label{cards}
If $\langle \kappa,\lambda,\theta,\vec W,\mathcal{S},\mathcal{T} \rangle$ is nice and $\mathcal{S}$ is stationary, then $\mathbb P^{\mathrm{dec}*}_{\kappa,\mathcal{S},\mathcal{T}}$ forces that $\lambda = \kappa^+$ and $\theta = \lambda^+$.
\end{lemma}

\begin{proof}
Since $\mathbb P = \mathbb P^{\mathrm{dec}*}_{\kappa,\mathcal{S},\mathcal{T}}$ is strongly proper for $\mathcal{S} \cup \mathcal{T}$, the regularity of $\lambda$ and $\theta$ are preserved.  Since $\mathcal{S}$ is stationary, for every $p \in \mathbb P$, and every $\alpha <\lambda$, there is $M \in \mathcal{S}$ such that $\alpha,p \in M$.  Let $q \leq p$ be such that $M$ appears in $q$, $M$ has a successor $N$ in $q$, and $q$ forces that $N$ is the next model above $M$ appearing in the generic.  Then a density argument shows that $q$ forces that the union of the decorations attached to $M$ will be a surjection from $\kappa$ to $N$.  Thus $\mathbb P$ forces that $|\alpha| \leq \kappa$.
Now let $\alpha < \theta$ and $p \in \mathbb P$ be arbitrary.  Let $W \in \mathcal{T}$ be such that $\alpha,p \in W$.  Let $W'$ be the next transitive model above $W$.  Then by Lemma \ref{contchains}, $\mathbb P$ forces that $W'$ is the union of a $\subseteq$-increasing chain of sets of size $<\lambda$, so it forces that $|\alpha| \leq \lambda$.
\end{proof}

The remainder of this section borrows ideas from \cite{boban}.
Suppose $\langle \kappa,\lambda,\theta,\vec W,\mathcal{S},\mathcal{T} \rangle$ is nice, and let $\mathbb P$ be either $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}$ or $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}^{\mathrm{dec}*}$.  Suppose $G \subseteq \mathbb P$ is generic over $V$.
For notational convenience, let $W_0 = \emptyset$ and $W_\theta = H_\theta$.
Let us say $\delta \leq \theta$ is \emph{relevant} if either $\delta = 0$, $\delta = \theta$, or $W_\delta \in \mathcal{T}$.
We define a decreasing sequence of sets $\mathcal{S}_\delta^G$ for relevant $\delta \leq \theta$:
 $\mathcal{S}_\delta^G = \{ M \in \mathcal{S} : M \cap W_\delta$ appears in $G \cap W_\delta \}$.
Note that for $W_\delta \in \mathcal{T}$, $G \cap W_\delta$ is $(\mathbb P \cap W_\delta)$-generic over $V$ by strong properness, and $\mathcal{S}_\delta^G \in V[G\cap W_\delta]$.

\begin{lemma}
\label{statpres}
Suppose $G \subseteq \mathbb P$ is generic.  If $\delta$ is relevant and $\mathcal{S}' \subseteq \mathcal{S}_\delta^G$ is stationary in $V[G \cap W_\delta]$, then $\mathcal{S}' \cap \mathcal{S}^G_\theta$ is stationary in $V[G]$.
\end{lemma}

\begin{proof}
Suppose $G$, $\delta$, and $\mathcal{S}'$ are as hypothesized.  Let $F : W_\theta^{<\omega} \to W_\theta$ be a function in $V[G]$.  Work in $V[G \cap W_\delta]$.
Let $\dot F$ be a $(\mathbb P / (G \cap W_\delta))$-name for $F$, and let $p_0 \in \mathbb P / (G \cap W_\delta)$ be arbitrary.  Let $\theta^*>\theta$ be regular and let $\frak A$ be the structure $\langle H_{\theta^*},\in,\mathbb P,G\cap W_\delta,p_0,\dot F \rangle$ (as defined in $V[G \cap W_\delta]$).  Let $N \prec \frak A$ be such that $N \cap W_\theta = M \in \mathcal{S}'$.
Now go back to $V$.  Let $s$ be the set of models of $p_0$.  By Lemma \ref{spdec} (or \cite[Corollary 2.32]{neeman}), there is $p_1 \leq p_0$ in $\mathbb P$ in which $M$ appears, and if $t$ is the set of models of $p_1$, then $t$ is the closure of $s \cup \{M \}$ under intersections, and if we are using the decorated poset, then the decorating function is unchanged.
By hypothesis, $M \cap W_\delta$ appears in $G \cap W_\delta$, so $p_1 \restriction W_\delta \in G \cap W_\delta$, since $p_1 \restriction W_\delta$ is the weakest condition extending $p_0 \restriction W_\delta$ in which $M \cap W_\delta$ appears.
$p_1$ is a strong master condition for $M$, and thus it forces that $M$ is closed under $\dot F$.  By the arbitrariness of $p_0$, it is forced by $\mathbb P / (G \cap W_\delta)$ that there is a model in $\mathcal{S}' \cap \mathcal{S}_\theta^{\dot G}$ that is closed under $\dot F$.
\end{proof}

\begin{lemma}
\label{*proj}
Suppose $\langle \kappa,\lambda,\theta,\vec W,\mathcal{S},\mathcal{T} \rangle$ is nice and $\mathcal{S}$ is stationary.
 Assume also that
 $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}^{\mathrm{dec}*}$ preserves the regularity of $\kappa$.  If $G \subseteq \mathbb P_{\kappa,\mathcal{S},\mathcal{T}}^{\mathrm{dec}*}$ is generic over $V$, then in $V[G]$, $\ns_\lambda \restriction \cof(\kappa)$ is the canonical projection of $\ns \restriction \mathcal{S}^G_\theta$.  Equivalently, for every stationary $A \subseteq \lambda \cap \cof(\kappa)$ in $V[G]$, $\{ M \in \mathcal{S}^G_\theta : M \cap \lambda \in A \}$ is stationary.
\end{lemma}

\begin{proof}
Suppose $A \subseteq \lambda \cap \cof(\kappa)$ is a stationary set in $V[G]$.  Let $\delta$ be such that $W_\delta \in \mathcal{T}$ and $A \in V[G \cap W_\delta]$.  Let $\mathcal{S}^G_\delta(A) = \{ M \in \mathcal{S}^G_\delta : M \cap \lambda \in A \}$.  By Lemma \ref{statpres}, it suffices to show that $\mathcal{S}^G_\delta(A)$ is stationary in $V[G \cap W_\delta]$.
Note that for all $\eta$ such that $\delta<\eta<\theta$ and $W_\eta \in \mathcal{T}$, $\{ M \cap W_\eta : M \in \mathcal{S}^G_\delta(A)  \} = \mathcal{S}^G_\delta(A) \cap W_\eta$.  If $\delta<\eta<\eta'<\theta$ and $\mathcal{S}^G_\delta(A) \cap W_{\eta'}$ is a stationary subset of $\mathcal{P}_\lambda(W_{\eta'})$, then $\mathcal{S}^G_\delta(A) \cap W_{\eta}$ is a stationary subset of $\mathcal{P}_\lambda(W_{\eta})$.

Suppose towards a contradiction that $\mathcal{S}^G_\delta(A)$ is nonstationary in $V[G \cap W_\delta]$, and let $F : W_\theta^{<\omega} \to W_\theta$ be a function in $V[G \cap W_\delta]$ such that no $M \in \mathcal{S}^G_\delta(A)$ is closed under $F$.
There is $\eta$ such that $\delta<\eta<\theta$ and $W_\eta$ is closed under $F$, and thus $F \restriction W_\eta$ witnesses that $\mathcal{S}^G_\delta(A) \cap W_{\eta}$ is nonstationary in $\mathcal{P}(W_\eta)$.

 Hence $\mathcal{S}^G_\delta(A) \cap W_{\eta'}$ is nonstationary for $\eta\leq\eta'<\theta$.  Now let $W_{\eta'}$ be the successor of $W_\eta$ in $\mathcal{T}$.  By Lemma \ref{contchains}, the models in $(W_\eta,W_{\eta'})_G$ form a chain of length $\lambda$ that is $\in$- and $\subseteq$-increasing and continuous at points of cofinality $\kappa$.  In $V[G]$, let $\{ M_\alpha : \alpha < \lambda \}$ enumerate this chain.  Since $A$ is stationary in $V[G]$, $\{ \alpha \in A : M_\alpha \cap \lambda = \alpha \}$ is also stationary in $V[G]$.  But this means that $\mathcal{S}^G_\delta(A) \cap W_{\eta'}$ is stationary in $V[G]$, a contradiction.
\end{proof}

We will also use the following observation due to Veli\v{c}kovi\'c \cite[Claim 2.7]{boban}:

\begin{lemma}
\label{bobanclaim}
Suppose $\mathcal{S},\mathcal{T}$ are appropriate for $\kappa$ and $K$ and $\lambda$ is a cardinal such that each model $M \in \mathcal{S}$ has $M \cap \lambda \in \lambda$.  If $p$ is a condition in $\mathbb P_{\kappa,\mathcal{S},\mathcal{T}}$, $M$ and $N$ are in $p \cap \mathcal{S}$, and $M \cap \lambda < N \cap \lambda$, then $M \cap N \in N$.
\end{lemma}

\section{End-extending cardinals and $\ns_{\omega_1}$}
\label{sec:ee}
For the purposes of warm-up and motivation, we investigate in this section a relatively weak large cardinal notion and its forcing applications for the nonstationary ideal on $\omega_1$.

For two sets $M,N$ we say that $N$ is an end-extension of $M$, or $M \sqsubseteq N$, when $N \cap \rank(M) = M$.
Suppose $\kappa$ is a cardinal and $\mathcal{S} \subseteq [\kappa]^{<\kappa}$.  We say that $\mathcal{S}$ \emph{admits gap end-extensions} when for every structure $\frak A$ on $\kappa$ in a countable language, there is an expansion $\frak A^*$ of $\frak A$, also in a countable language, such that for every $M \in \mathcal{S}$ elementary in $\frak A^*$, there are cofinally many $\alpha<\kappa$ such that for some $N \prec \frak A^*$, $M \cup \{\alpha\} \subseteq N$ and $N \cap \alpha = M$.
For a cardinal $\mu\leq\kappa$, we say that $\kappa$ is $\mu$-end-extending when $[\kappa]^{<\mu}$ admits gap end-extensions, and we say $\kappa$ is end-extending when it is $\kappa$-end-extending.

\begin{lemma}
\label{statgap}
Suppose $\mu<\kappa$ and $\mathcal{S} \subseteq [\kappa]^{<\kappa}$ admits gap end-extensions.  Then for all structures $\frak A$ on $\kappa$ and all clubs $C \subseteq \kappa$, there is an expansion $\frak A^*$ such that for all $M \prec \frak A^*$ with $M \in \mathcal{S}$ and $M \cap \mu \in \mu$, there are cofinally many $\alpha \in C$ such that $\cf(\alpha)\geq\mu$ and $\sk^{\frak A^*}(M \cup \{\alpha\}) \cap \alpha = M$.
\end{lemma}

\begin{proof}
Let $f : \kappa^2 \to \kappa$ be such that if $\cf(\alpha) \leq \beta$, then $\{ f(\alpha,\gamma) : \gamma <\beta \}$ is cofinal in $\alpha$.
Let $\frak A$ be any structure on $\kappa$ with $f$, $C$, and $\mu$ in its language.  Then for $\alpha<\kappa$, $\cf(\alpha)$ is definable in $\frak A$ as the least $\beta$ such that for all $\gamma<\alpha$, there is $\delta<\beta$ with $f(\alpha,\delta) \geq \gamma$.  Let $\frak A^*$ be an expansion witnessing that $\mathcal{S}$ admits gap end-extensions.  Suppose $M \prec \frak A^*$ is in $\mathcal{S}$ and $M \cap \mu \in \mu$.  For $\alpha<\kappa$, let $N_\alpha = \sk^{\frak A^*}(M \cup \{\alpha \})$.  If $\alpha > \sup(M)+1$ is such that $N_\alpha \cap \alpha = M$, then $\cf(\alpha) \geq \mu$, because otherwise, $\cf(\alpha) \in N_\alpha \cap \mu$ and $\{ f(\alpha,\delta) : \delta < \cf(\alpha) \}$ is cofinal in $\alpha$ and contained in $N_\alpha$.  Also, if $N_\alpha \cap \alpha = M$, then $N_\alpha \models$ ``$C$ is unbounded in $\alpha$,'' so $\alpha \in C$ by elementarity.
\end{proof}

\begin{prop}
If $\mu$ is regular, $\mu<\kappa$, and $\kappa$ is $\mu$-end-extending, then $\cf(\kappa)>\mu$.
If $\kappa$ is end-extending, then $\kappa$ is weakly inaccessible.
\end{prop}

\begin{proof}
Suppose $\mu$ is regular, $\mu<\kappa$, and $\kappa$ is $\mu$-end-extending.  Towards a contradiction, suppose that $\cf(\kappa) \leq \mu$.  Let $C \subseteq \kappa$ be a club consisting of ordinals of cofinality $<\mu$.
By the previous lemma, there is a set $M \in \mathcal{P}_\mu(\kappa)$ and an $\alpha \in C$ above $\sup(M)$ such that $\cf(\alpha) \geq \mu$.  This is a contradiction.

Suppose $\kappa$ is end-extending.  Then it is $\mu$-end-extending for all $\mu<\kappa$, so by the previous paragraph, $\kappa$ is regular.  Note that $\kappa>\omega_1$, since we can take $\frak A$ on $\omega_1$ such that all substructures of $\frak A$ are transitive, while admitting gap end-extensions requires non-transitive substructures.
If $\kappa = \nu^+$, let $f(x,y)$ be a function such that for all $\alpha<\kappa$, $f(\alpha,\cdot)$ is an injection of $\alpha$ into $\nu$.  Let $\frak A$ incorporate $f$ in its language, and let $\frak A^*$ be given by the hypothesis.  Let $M \prec \frak A^*$ be such that $\nu \in M$ and $| M \cap \nu | < \nu$.  We can inductively build $N \prec \frak A^*$ such that $N$ is cofinal in $\kappa$ and $M \sqsubseteq N$, by applying end-extendibility at successor stages and taking unions at limits.  Let $\alpha\in N$ be such that $|N \cap \alpha| \geq \nu$.  Then $f(\alpha,\cdot)$ injects $N \cap \alpha$ into $N \cap \nu$, which is a contradiction since $N \cap \nu = M \cap \nu \in [\nu]^{<\nu}$.
\end{proof}

\begin{prop}
Suppose $\kappa$ is $\mu$-end-extending.
\begin{enumerate}
\item For all infinite cardinals $\nu<\delta<\kappa$, $(\kappa,\delta) \twoheadrightarrow (\mu,\nu)$.
\item $0^\sharp$ exists.
\item If $\mu = \kappa$, then $\kappa$ is Rowbottom and weakly $\kappa$-Mahlo.
\end{enumerate}
\end{prop}

\begin{proof}

For the first item, let $\frak A$ be any structure on $\kappa$ and let $\frak A^*$ be an expansion witnessing that $\kappa$ is $\mu$-end-extending.  Let $M \prec \frak A^*$ be such that $\delta \in M$ and $|M \cap \delta| = \nu$.  Build a strictly $\sqsubseteq$-increasing continuous sequence $\langle M_i : i \leq \mu \rangle$ with $M_0 = M$.  Then $M_\mu \prec \frak A$, $|M_\mu \cap \kappa| = \mu$, and $|M_\mu \cap \delta| = \nu$.

The second item follows from $(\kappa,\delta) \twoheadrightarrow (\mu,\nu)$, using Theorem 18.27 and the argument for Corollary 18.29 in \cite{jech}.

For the third item, let $c$ be any coloring of the finite subsets of $\kappa$ in $\lambda$-many colors, $\lambda<\kappa$, and let $\frak A$ be a structure incorporating $c$ into its language, such that every submodel of size $<\kappa$ can be end-extended.  By repeatedly end-extending a countable $M \prec \frak A$ with $\lambda \in M$, we obtain a subset of $\kappa$ of size $\kappa$ on which $c$ takes only countably many values.  This implies that $\kappa$ is a regular Rowbottom cardinal, and by Shelah \cite{shelah}, that it is weakly $\kappa$-Mahlo.
\end{proof}

The following is inspired by arguments in \cite{cox} and \cite{sakai}.

\begin{prop}
Suppose $\ensuremath{\mathcal{I}}$ is a $\kappa$-complete ideal on $\kappa$, $\mathcal{S}^* \subseteq [H_{(2^{2^\kappa})^+}]^{<\kappa}$ is stationary, $\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}$ is $\mathcal{S}^*$-proper, and $\mathcal{S} = \{ M \cap\kappa : M \in \mathcal{S}^* \}$.  Then $\mathcal{S}$ admits gap end-extensions.
\end{prop}

\begin{proof}
First we claim that $\ensuremath{\mathcal{I}}$ is precipitous.  Let $X$ be an $\ensuremath{\mathcal{I}}$-positive set, and suppose $X \Vdash$ ``$\vec\tau = \langle \tau_n : n \in \omega \rangle$ is a descending sequence of ordinals in the generic ultrapower.''  Let $\mathcal{D}_n$ be the dense set of conditions deciding $\tau_n = \check f$ for some function $f$ on $\kappa$.
Let $\frak A = \langle H_{(2^{2^\kappa})^+},\in,\lhd,\ensuremath{\mathcal{I}},X,\vec\tau \rangle$, where $\lhd$ is a well-order, and let $M \prec \frak A$ be in $\mathcal{S}^*$.  Let $Y \subseteq X$ be $(M,\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}})$-generic.

Note that for each dense set $\mathcal{D} \subseteq \mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}$ in $M$, $Y \setminus \bigcup (M \cap \mathcal{D}) \in \ensuremath{\mathcal{I}}$.  Otherwise, there is an $\ensuremath{\mathcal{I}}$-positive $Z \subseteq Y$ such that $Z \cap A = \emptyset$ for each $A \in \mathcal{D} \cap M$.  But this contradicts that $Z$ is $(M,\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}})$-generic.
By $\kappa$-completeness, there is $\alpha \in Y$ such that for each dense $\mathcal{D} \in M$, there is $A \in \mathcal{D} \cap M$ with $\alpha \in A$.  Let $\mathcal{U} = \{ A \in \mathcal{P}(\kappa) \cap N : \alpha \in A \}$.  Then $\mathcal{U}$ is an $(M,\mathcal{P}(\kappa)/I)$-generic ultrafilter containing $X$.  For each $n$, let $f_n$ be a function such that some $A \in \mathcal{D}_n \cap \mathcal{U}$ decides $\tau_n = \check f_n$.  By the statement forced by $X$, $\{ \beta : f_{n+1}(\beta) \in f_n(\beta) \} \in \mathcal{U}$ for each $n$.  So $\langle f_n(\alpha) : n \in \omega \rangle$ is a descending sequence of ordinals, a contradiction.

Now let $\frak A$ be a structure on $\kappa$ let $\xi<\kappa$.  Let $\frak B = \langle H_{(2^{2^\kappa})^+},\in,\lhd,\ensuremath{\mathcal{I}},\frak A \rangle$, where $\lhd$ is a well-order.  Let $\frak A^*$ be the result of restricting the Skolem functions of $\frak B$ to $\kappa$.
Let $N^* \in \mathcal{S}^*$ be elementary in $\frak B$, and let $G$ be a $\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}$-generic filter with a master condition for $N^*$.  Let $N=N^*\cap\kappa\prec\frak A^*$.

Let $j : V \to M$ be the generic ultrapower embedding.  Since $|N^*|<\kappa$, $j(N^*) = j[N^*]$.  Suppose $\alpha$ is an ordinal in $Q = \sk^{j(\frak B)}(j(N^*) \cup \{\kappa \})$.  Then there is a function $f : \kappa \to \ord$ in $N^*$ such that $\alpha = j(f)(\kappa)$.  Let $\tau \in N^*$ be a $\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}$-name for $j_{\dot{G}}(f)(\kappa)$.  Since $G$ has a master condition for $N^*$, $\tau^G \in N^*$.  Since $j(N^*) \cap \kappa = N^* \cap \kappa = N$, we have $Q \cap \kappa = N$.  By elementarity, there is $\beta$ such that $\xi<\beta<\kappa$ and $\sk^{\frak A^*}(N \cup \{\beta\}) \cap \beta = N$.
\end{proof}

It follows that measurable cardinals are end-extending.  But since $\kappa$ being end-extending depends only on $\mathcal{P}(\kappa)$, reflection shows that every normal ultrafilter on $\kappa$ has a measure-one set of end-extending cardinals.
  Furthermore, since we can consistently have weakly inaccessible cardinals carrying $\omega_1$-saturated ideals, end-extending cardinals are not necessarily strong limit.  Even successor cardinals can be end-extending in limited degrees.  For example, the statement that $\omega_2$ is $\omega_1$-end-extending is easily seen to be equivalent to the notion $\mathrm{SCC}^{\cof}_{\mathrm{gap}}$ defined in \cite{cox}.    It follows from the above proposition that if $\kappa$ carries a $\kappa$-complete ideal $\ensuremath{\mathcal{I}}$ such that $\mathcal{P}(\kappa)/\ensuremath{\mathcal{I}}$ is a proper forcing, then $\kappa$ is $\omega_1$-end-extending.  If $\kappa$ is measurable and $\mu<\kappa$ is regular, then this situation is forced the L\'evy collapse $\col(\mu,{<}\kappa)$ (see \cite{foreman}).

Let us now see how this notion can be used with Neeman's forcing.  Suppose $\theta^{<\theta} = \theta$ and $\alpha^\omega<\theta$ for each $\alpha<\theta$.
Let $\vec W$ be a filtration of $H_\theta$, and let $\frak A = \langle H_\theta,\in,\vec W \rangle$.  Let $\mathcal{S}$ be the set of countable $M \prec \frak A$, and let $\mathcal{T}$ be the set of $W_\alpha \prec \frak A$ such that $\cf(\alpha)$ is uncountable.  Then according to the terminology of the previous section, $\langle \omega,\omega_1,\theta,\vec W,\mathcal{S},\mathcal{T} \rangle$ is nice.  Let us set $\mathbb C_{\vec W} = \mathbb P^{\mathrm{dec}*}_{\omega,\mathcal{S},\mathcal{T}}$ under these hypotheses.
If $\theta$ is inaccessible, let $\mathbb C_\theta =\mathbb C_{\vec W}$ for $\vec W = \langle V_\alpha : \alpha<\theta\rangle$.

\begin{theorem}
\label{wps-ee}
If $\theta$ is an inaccessible $\omega_1$-end-extending cardinal, or a stationary limit of such cardinals, then $\mathbb C_\theta$ forces that $\ns_{\omega_1}$ is weakly presaturated.  If $\theta^{<\theta} = \theta$, $\alpha^\omega<\theta$ for all $\alpha<\theta$, and $\theta$ is $\omega_1$-end-extending, then the same conclusion is forced by $\mathbb C_{\vec W}$, where $\vec W$ is any filtration of $H_\theta$.
\end{theorem}

\begin{proof}
Let us first show the latter statement.
Let $G \subseteq \mathbb C_{\vec W} = \mathbb P^{\mathrm{dec}*}_{\omega,\mathcal{S},\mathcal{T}}$ be generic, and let $\mathcal{S}^G_\theta$ be the collection of $M \in \mathcal{S}$ that appear in $G$.  By Lemma \ref{cards}, $\mathbb P$ forces $\theta$ to become $\omega_2$.
By Lemma \ref{*proj}, $V[G]$ satisfies that $\ns_{\omega_1}$ is the canonical projection of $\ns \restriction \mathcal{S}^G_\theta$.
By Lemma \ref{wps-equivs}, it suffices to show that for all stationary $A \subseteq \omega_1$ and $f : \omega_1 \to \omega_1$, $\{ M \in \mathcal{S}^G_\theta : M \cap \omega_1 \in A \wedge f(M \cap \omega_1) <\ot(M \cap \theta) \}$ is stationary.

Let $\dot A$ be a name for a stationary subset of $\omega_1$, $\dot f$ a name for a function from $\omega_1$ to itself, and $\dot F$ a name for a function from $(H_\theta^V)^{<\omega}$ to $H_\theta^V$.  Let $p_0 \in \mathbb C_{\vec W}$ be arbitrary.  Let $\lambda > \theta$ be regular and let $\frak B = \langle H_\lambda,\in,\frak A,p_0,\dot f,\dot F,\lhd \rangle$, where $\lhd$ is a well-order.  Let $\frak A^*$ be the result of restricting the Skolem functions for $\frak B$ to $H_\theta$.  Let $\frak A^{**}$ be an expansion of $\frak A^*$ given by the $\omega_1$-end-extending hypothesis.

In $V[G]$, there is some $M \prec \frak A^{**}$ such that $M \in S^G_\theta$ and $M \cap \omega_1 \in A$.  Let $p_1 \leq p_0$ decide the value of such $M$, force that $M \cap \omega_1 \in \dot A$, and decide $\dot f(M \cap \omega_1) = \xi$ for some ordinal $\xi<\omega_1$.  We may assume that $p_1$ takes the form $\langle s_1,g \rangle$ with $M \in s_1$.

Let $\alpha$ be least such that $p_1 \in W_\alpha\in\mathcal{T}$.  By end-extendibility, there is $M_1 \prec \frak A^{**}$ such that $M \sqsubseteq M_1$, and if $\alpha_0$ is the least ordinal in $M_1 \setminus M$, then $\alpha\leq\alpha_0$.
By Lemma \ref{rankmeet}, $W_{\alpha_0} \in \mathcal{T}$.  Now repeat this $(\xi+1)$-many times to obtain a continuous $\subseteq$-increasing sequence of models $\langle M_i : i \leq \xi+1 \rangle$, all elementary in $\frak A^{**}$, with $M_0 = M$, and a corresponding sequence of ordinals $\langle \alpha_i : i \leq \xi \rangle$ such that $\alpha_i \in M_{i+1}$, $M_{i+1} \cap W_{\alpha_i} = M_i$.  Let $M^* = M_{\xi+1}$.  Then $M^* \cap W_{\alpha_0} = M$, and $\ot(M^* \cap \theta) > \xi$.

Let $s_2 = s_1 \cup \{ W_{\alpha_0}, M^* \}$.  Clearly, $s_2$ is an $\in$-chain.  To see that it is closed under intersections, note that $W_{\alpha_0} \cap M^* = M \in s_1$, and for any $Q \in s_1$, $Q \cap W_{\alpha_0} = Q$ and $M^* \cap Q = M^* \cap W_{\alpha_0} \cap Q = M \cap Q \in s_1$.
 Then $p_2 = \langle s_2,g \rangle$ is a condition below $p_1$.

Let $N^* = \sk^{\frak B}(M^*)$.  Since $p_2$ is a master condition for $N^*$ and $\dot F \in N^*$,  $p_2$ forces that $N^*[G]\cap H_\theta^V = M^*$, so it forces $M^*$ to be closed under $\dot F$.  Also, $p_2$ forces $M^*$ to be in $\mathcal{S}^G_\theta$ and $M \cap \omega_1 = M^* \cap \omega_1 \in \dot A$.  Since $p_0$ and $\dot F$ were arbitrary, it is forced that the set
$\{ Q \in \mathcal{S}^G_\theta : Q \cap \omega_1 \in A \wedge f(Q \cap \omega_1) < \ot(Q \cap \theta) \}$
is stationary.

Now suppose that $\theta$ is an inaccessible stationary limit of regular $\omega_1$-end-extending cardinals.
Let $\dot A,\dot f,\dot F,p_0$ be as above.
By the chain condition, we may assume that $\dot A,\dot f,p_0 \in V_\theta$, and on a club of $\alpha<\theta$, $\dot F \cap V_\alpha$ is forced to be a name for $\dot F \restriction V_\alpha$.  We can find an inaccessible $\kappa<\theta$ such that $\kappa$ is $\omega_1$-end-extending, $\dot A,\dot f,p_0 \in V_\kappa \in \mathcal{T}$, and $\dot F \cap V_\kappa$ is a $\mathbb C_\kappa$-name for a function from $V_\kappa^{<\omega}$ to $V_\kappa$.  By the arguments of the previous paragraphs, there is $q \leq p_0$ in $V_\kappa$ that forces some $M \in \mathcal{S} \cap V_\kappa$ to appear in the generic $G \cap V_\kappa$, be closed under $\dot F$, and have the properties that  $M \cap \omega_1 \in \dot A$ and  $\dot f(M \cap \omega_1) < \ot(M \cap \kappa)$.  Since $V_\kappa \in \mathcal{T}$,
$\mathbb C_\kappa$ is a regular suborder of $\mathbb C_\theta$,
and any generic $G \subseteq \mathbb C_\theta$ possessing $q$ will yield an extension satisfying these statements.  By the arbitrariness of $p_0$ and $\dot F$, the collection of such models $M$ is forced to be stationary.
\end{proof}

We remark that above forcing does not necessarily render $\ns_{\omega_1}$ precipitous.  Claverie and Schindler \cite{clav} showed that a precipitous weakly presaturated ideal (a.k.a.\ ``strong ideal'') is equiconsistent with a Woodin cardinal.

Another strengthening of weak presaturation for $\ns_{\omega_1}$ is the statement that for every $f : \omega_1 \to \omega_1$, there is a club $C \subseteq \omega_1$ and a canonical function $\psi_\alpha$ such that $f(\beta) < \psi_\alpha(\beta)$ for all $\beta \in C$.
Deiser-Donder \cite{dd} and Larson-Shelah \cite{ls} showed that this property, which the latter call ``Bounding'', is equiconsistent with an inaccessible limit of measurable cardinals.

Besides consistency strength considerations, there is a combinatorial reason why $\mathbb C^*_\theta$ does not necessarily force Bounding.
Suppose $\Diamond_{\omega_1}$ holds in $V$.  Let $\langle x_\alpha : \alpha < \omega_1 \rangle$ be a diamond sequence.  Define a function $f : \omega_1 \to \omega_1$ as follows.  If $x_\alpha$ codes, via the G\"odel pairing function, a countable transitive set $N$, let $f(\alpha) = \ot(N \cap \omega_1)$.  Otherwise, let $f(\alpha) = 0$.

Let $\theta > \omega_1$ be regular and let $\frak A$ be a structure on $H_\theta$ in a countable language.
Let $\langle M_i : i < \omega_1 \rangle$ be a continuous, $\in$-increasing sequence of countable elementary substructures of $\frak A$.
Let $C \subseteq \omega_1$ be the club of $\alpha$ such that $M_\alpha \cap \omega_1 = \alpha$.
Let $M$ be the union of the $M_i$, and let $\bar M$ be the transitive collapse of $M$.
Let $A \subseteq \omega_1$ code the $\in$-structure of $\overline M$ via the pairing function.
There is a club $D \subseteq C$ such that for all $\alpha \in D$, $A \cap \alpha$ codes a structure isomorphic to $M_\alpha$.  There is some $\alpha \in D$ such that $x_{\alpha} = A \cap \alpha$.
We have $f(\alpha) = f(M_\alpha \cap \omega_1) =  \ot(M_{\alpha} \cap \theta)$.
Thus there are stationary-many $z \in \mathcal{P}_{\omega_1}(\theta)$ such that $f(z \cap \omega_1) = \ot(z \cap \theta)$.

If $\mathbb P$ is a proper forcing, then this continues to hold in $V^{\mathbb P}$ with respect to the same $f$.  In particular, it holds in any extension by $\mathbb C_\theta$.
Let $G \subseteq \mathbb P$ be generic.  There is some $M \prec H_{\omega_2}^{V[G]}$ such that $f \in M$, and $f(M \cap \omega_1) = \ot(M \cap \omega_2)$.
If Bounding were to hold in $V[G]$, then by elementarity, there would be an ordinal $\gamma \in M$ and a club $C \in M$ such that $M \models$ ``$f(\beta) < \psi_\gamma(\beta)$ for all $\beta \in C$''.  But $\alpha = M \cap \omega_1 \in C$ and $\psi_\gamma(\alpha) = \ot(M \cap \gamma) < \ot(M \cap \omega_2) = f(\alpha)$, a contradiction.

\section{$\wcc(\omega_2,\cof(\omega_1))$ from $(+2)$-subcompactness}
\label{sec:wccomega_2}

As far as we are able to ascertain, until the time of this writing, the best known upper bounds for the consistency strength of $\wcc(\mu^+,\cof(\mu))$, for $\mu$ regular uncountable, remained what was discovered in the late 1970s, namely an almost-huge cardinal.  Kunen \cite{kunen} developed a forcing strategy that collapses a huge cardinal $\kappa$ to become the successor of a chosen regular cardinal $\mu<\kappa$, and forces Chang's Conjecture $(\kappa^+,\kappa) \twoheadrightarrow (\mu^+,\mu)$ and the existence of a saturated ideal on $\kappa$.  Shortly thereafter, Magidor showed that the saturated ideal, and consequently $\wcc(\kappa,\cof(\mu))$, can be obtained from an almost-huge $\kappa$ (see \cite{foreman}).  In \cite{eh}, the author and Hayut showed by a reflection argument that the strength of $(\omega_3,\omega_2) \twoheadrightarrow (\omega_2,\omega_1)$ is weaker than a huge cardinal, and that a single huge cardinal can be used to get Chang's Conjecture between many pairs of cardinals simultaneously, but the reflection did not take us down to the level of almost-huge or lower.  On the other hand, we showed that if we allow some distance between the pairs of cardinals, the strength can be shown to be much lower by a very different forcing argument.  In particular, we got a model of GCH plus $(\omega_4,\omega_3) \twoheadrightarrow (\omega_2,\omega_1)$ from a model of GCH with a $(+2)$-subcompact cardinal.  We get some more information using an observation of Adolf \cite{adolf}:

\begin{lemma}[Adolf]
Suppose $2^{\kappa^+} = \kappa^{++}$ and $(\kappa^{+++},\kappa^{++}) \twoheadrightarrow (\kappa^+,\kappa)$.  Then for every regular $\mu\leq\kappa$ the set of $M \prec H_{\kappa^{+3}}$ such that $M \cap \kappa^+ \in \kappa^+$, $\cf(M \cap \kappa^{+}) = \mu$, $\cf(M \cap \kappa^{++}) = \kappa$, and $\ot(M \cap \kappa^{+3}) = \kappa^+$, is stationary.
\end{lemma}

Let us show that the above result yields many witnesses to Chang's Conjecture that are countably closed.  Suppose GCH holds and $M \prec H_{\omega_4}$ is such that $\omega_1 \subseteq M$ and $\cf(M \cap \kappa)>\omega$ for each uncountable cardinal $\kappa \in M$.  By CH, $M$ contains all reals.  Since $\cf(M \cap \omega_2)$ is uncountable, $[\omega_2]^{<\omega_1} \cap M = \bigcup \{ [\alpha]^{<\omega_1} : \alpha \in M \cap \omega_2 \} = [M \cap \omega_2]^{<\omega_1}$.  Similarly, $[\omega_3]^{<\omega_1} \cap M = \bigcup \{ [\alpha]^{<\omega_1} \cap M : \alpha \in M \cap \omega_3 \}$, and for any $\alpha \in [\omega_2,\omega_3) \cap M$, a bijection $f : \omega_2 \to \alpha$ in $M$ yields that $[\alpha]^{<\omega_1} \cap M = \{ f[z] : z \in [M \cap \omega_2]^{<\omega_1} \} = [M \cap \alpha]^{<\omega_1}$.  Thus $[\omega_3]^{<\omega_1} \cap M = [M \cap \omega_3]^{<\omega_1}$.  Similarly, $[\omega_4]^{<\omega_1} \cap M = [M \cap \omega_4]^{<\omega_1}$.  Thus $M$ is countably closed.

Now assume GCH and $(\omega_4,\omega_3) \twoheadrightarrow (\omega_2,\omega_1)$.  Let us introduce a Neeman forcing.  Let $\vec W$ be a filtration of $H_{\omega_4}$, and let $\frak A = \langle H_{\omega_4},\in,\vec W \rangle$.  Let $\mathcal{S}$ be the collection of all countably closed $N \prec \frak A$ of size $\omega_1$ such that $N \sqsubseteq M$ for some countably closed $M \prec \frak A$ with $\ot(M \cap \omega_4) = \omega_2$.  By taking countably closed elementary initial segments of witnesses to Chang's Conjecture, we see that $\mathcal{S}$ is stationary.  Let $\mathcal{T}$ be the collection of all $W_\alpha \prec \frak A$ with $\cf(\alpha) \geq \omega_2$.  Each $W \in \mathcal{T}$ is closed under $\omega_1$-sequences.  For any $M \in \mathcal{S}$ and $W \in \mathcal{T}$, $M \cap W$ is in $\mathcal{S}$, since $W \cap M \sqsubseteq M$, and it is a countably closed elementary substructure of $\frak A$.  Thus $\langle \mathcal{S},\mathcal{T} \rangle$ is appropriate for $\omega_1$ and $\frak A$.

Let $\mathbb P = \mathbb P^{\mathrm{dec}*}_{\omega_1,\mathcal{S},\mathcal{T}}$.  Then $\mathbb P$ is countably closed, preserves $\omega_2$ and $\omega_4$, and collapses $\omega_3$ so that $\omega_3^{V^{\mathbb P}} = \omega_4^V$.  Let $\dot F$ be a $\mathbb P$-name for a function from $(H_{\omega_4}^V)^{<\omega}$ to $H_{\omega_4}^V$.  Let $p_0 \in \mathbb P$ be arbitrary, and let $\dot f$ be a $\mathbb P$-name for a function from $\omega_2$ to $\omega_2$.
Let $\frak B = \langle H_{\omega_5},\in,\lhd,\vec W,\dot F,p_0,\dot f \rangle$, where $\lhd$ is a well-order.  Let $\frak B^*$ be the result of restricting the Skolem functions for $\frak B$ to $H_{\omega_4}$.

By $(\omega_4,\omega_3) \twoheadrightarrow (\omega_2,\omega_1)$ and the above lemma, there are stationary-many countably closed $M \prec \frak B^*$ such that $\ot(M \cap \omega_4) = \omega_2$ and $|M \cap \omega_3| = \omega_1$.  For each $\alpha < \omega_4$, let $M_\alpha \prec \frak B^*$ be such a model with $\alpha \in M_\alpha$.  For each $\alpha<\omega_4$ of cofinality at least $\omega_2$, $M_\alpha \cap \alpha$ is bounded below $\alpha$.  Using Fodor's Lemma, there is a stationary $A \subseteq \omega_4 \cap \cof(\geq\omega_2)$ and a $\gamma<\omega_4$ such that $M_\alpha \cap \alpha \subseteq \gamma$ for all $\alpha \in A$.  By GCH, there is a stationary $B \subseteq A$ and a set $N$ such that $M_\alpha \cap W_\alpha = N$ for all $\alpha \in B$.  We may assume that $W_\alpha \prec \frak B^*$ for all $\alpha \in B$, which implies $N \prec \frak B^*$.  Note that $N \in \mathcal{S}$.

Let $p_1 = \langle s_1,h \rangle$ be a condition below $p_0$ with $N \in s_1$, such that $p_1$ decides the value of $\dot f(N \cap \omega_2)$, say as $\xi<\omega_2$.  Let $\alpha \in B$ be such that $p_1 \in W_\alpha$.  Let $N^* \sqsubseteq M_\alpha$ be such that $N^* \prec M_\alpha$, $N^*$ is countably closed, and $\ot(N^* \cap \omega_4) > \xi$.  Let $s_2 = s_1 \cup \{ W_\alpha,N^* \}$.  Then $s_2$ is closed under intersections, since $N^* \cap W_\alpha = N \in s_1$, and for any $Q \in s_1$, $W_\alpha \cap Q = Q$ and $N^* \cap Q = N^* \cap W_\alpha \cap Q = N \cap Q \in s_1$.
Then $p_2 = \langle s_2,h \rangle$ is a condition below $p_1$, and it forces that $N^*$ is closed under $\dot F$ and that $\dot f(N^* \cap \omega_2)< \ot(N^* \cap \omega_2)$.  Note also that $\cf(N^* \cap \omega_2) = \omega_1$.  As $p_0,\dot f$, and $\dot F$ were arbitrary, we have:
\begin{theorem}
If ZFC+GCH is consistent with a (+2)-subcompact cardinal, then ZFC is consistent with $\wcc(\omega_2,\cof(\omega_1))$.
\end{theorem}

\section{Magidor models and ambitious cardinals}
\label{sec:ambitious}

Suppose $\kappa$ is a regular cardinal.  Following \cite{mv}, we call a set $M$ \emph{$\kappa$-Magidor} if $\kappa \in M$, $M \cap \kappa \in \kappa$, $M$ is extensional, $M$ has size ${<}\kappa$, and the transitive collapse of $M$ is equal to $V_\alpha$ for some $\alpha<\kappa$.  Magidor \cite{magidor} showed that $\kappa$ is supercompact if and only if for every $\theta > \kappa$, the set of $\kappa$-Magidor $M \in \mathcal{P}_\kappa(V_\theta)$ is stationary.

\begin{lemma}
\label{magclosure}
Suppose $\mu<\kappa$ is a regular cardinal, $M \prec V_\theta$ is a $\kappa$-Magidor model, $\mu \leq M \cap \kappa$, and $\cf(\sup(M \cap \theta)) \geq \mu$.  Then $M$ is ${<}\mu$-closed.
\end{lemma}

\begin{proof}
Let $i : V_\eta \to M$ be the inverse of the transitive collapse of $M$. Suppose $x \in [M]^{<\mu}$.  Since $\cf(M \cap \theta)\geq\mu$, there is $\alpha < \eta$ such that $x \subseteq i(V_\alpha)$.  Then $y = i^{-1}[x] \in V_\eta$, and $x = i(y) \in M$ since $|y|<M\cap\kappa=\crit(i)$.
\end{proof}

In this section, we identify a species of supercompactness that has a characterization in terms of Magidor models that is somewhat analogous to how countable models behave with end-extending cardinals.

\begin{definition*}
A cardinal $\kappa$ is \emph{ambitious} when for all $\lambda\geq\kappa$, there is $\delta_0 \geq \lambda$ such that for all $\delta \geq \delta_0$, there is a $\lambda$-closed transitive $M$, a $\delta$-closed transitive $N$, and elementary embeddings $j : V \to M$ and $k : M \to N$
such that:
\begin{itemize}
\item $\crit(j) = \kappa$ and $\lambda < j(\kappa)$.
\item $\crit(k)>\lambda$, $k[j(\kappa)] \subseteq \delta$, and $k(j(\kappa)) > \delta$.
\end{itemize}
\end{definition*}

Recall that $\kappa$ is \emph{almost-huge} if there is an elementary $j : V \to M$ with critical point $\kappa$ and $M$ is a transitive class closed under ${<}j(\kappa)$-sequences.
The name \emph{ambitious} is chosen because it is as if such $\kappa$ are trying to be almost-huge.  The first embedding $j$ sends $\kappa$ to a highly closed model, but $j(\kappa)$ overshoots the closure.  The next embedding $k$ tries to make up for this by sending $j(\kappa)$ into a model whose closure is above $k(\alpha)$ for each $\alpha$ at which $j$ may have fallen short of being an almost-huge embedding.  Of course, $k(j(\kappa))$ may overshoot the new closure, but it looks like progress.  Indeed, this is a local property of systems of measures that characterize almost-huge cardinals \cite[Theorem 24.11]{kanamori}.

\begin{prop}
The following are equivalent:
\begin{enumerate}
\item\label{amb} $\kappa$ is ambitious.

\item\label{amb-uf} (High-jump property) For all $\lambda \geq \kappa$, there is $\delta_0 \geq \lambda$ such that for all $\delta \geq \delta_0$, there is a normal ultrafilter $\mathcal{U}$ on $\mathcal{P}_\kappa(\delta)$ such that for all $f : \mathcal{P}_\kappa(\lambda) \to \kappa$, $\{ z \in \mathcal{P}_\kappa(\delta) : f(z \cap \lambda) < \ot(z) \} \in \mathcal{U}$.

\item\label{amb-uf-ptwise} (Shelah property) For all $\lambda \geq \kappa$ and all $f : \mathcal{P}_\kappa(\lambda) \to \kappa$, there is $\delta_0 \geq \lambda$ such that for all $\delta \geq \delta_0$, there is a normal ultrafilter $\mathcal{U}$ on $\mathcal{P}_\kappa(\delta)$ such that $\{ z \in \mathcal{P}_\kappa(\delta) : f(z \cap \lambda) < \ot(z) \} \in \mathcal{U}$.

\end{enumerate}
\end{prop}

\begin{proof}
(\ref{amb}) $\Rightarrow$ (\ref{amb-uf}): Let $\lambda\geq\kappa$, let $\delta_0 \geq \lambda$ be given by (\ref{amb}), let $\delta\geq\delta_0$, and let $j : V \to M$ and $k : M \to N$
also be given by (\ref{amb}).  Let $i = k \circ j$.  Let $\mathcal{U} = \{ X \subseteq \mathcal{P}_\kappa(\lambda) : j[\lambda] \in j(X) \}$, and let $\mathcal{W} =  \{ X \subseteq \mathcal{P}_\kappa(\delta) : i[\delta] \in i(X) \}$.  Let $\pi : \mathcal{P}_\kappa(\delta) \to \mathcal{P}_\kappa(\lambda)$ be $z \mapsto z \cap \lambda$.  Then:
\begin{align*}
X \in \mathcal{U} &\Leftrightarrow j[\lambda] \in j(X) \\
		&\Leftrightarrow k(j[\lambda]) = i[\lambda] \in i(X) \text{\hspace{5mm}(since $\crit(k) > \lambda)$} \\
		&\Leftrightarrow i(\pi)(i[\delta]) \in i(X) \\
		&\Leftrightarrow \pi^{-1}[X] \in \mathcal{W}
\end{align*}
Let $\bar k : \Ult(V,\mathcal{U}) \to \Ult(V,\mathcal{W})$ be the map $[f]_\mathcal{U} \mapsto [f \circ \pi]_\mathcal{W}$.
Let $\ell_\mathcal{U} : \Ult(V,\mathcal{U}) \to M$ be $[f]_{\mathcal{U}} \mapsto j(f)(j[\lambda])$, and let $\ell_\mathcal{W} : \Ult(V,\mathcal{W}) \to N$ be $[f]_{\mathcal{W}} \mapsto i(f)(i[\delta])$.
With $j_\mathcal{U},j_\mathcal{W}$ denoting the usual ultrapower embeddings, we have the following commutative diagram:

\begin{diagram}
V			&			&\rTo^{j}				&&M					&\rTo^{k}		&N			\\
			&\rdTo^{j_\mathcal{U}} \rdTo(4,2)^{j_\mathcal{W}}		&&\ruTo^{\ell_\mathcal{U}}	&			&\ruTo^{\ell_\mathcal{W}}	\\
			&				&\Ult(V,\mathcal{U})		&\rTo^{\bar k}	&\Ult(V,\mathcal{W})
\end{diagram}
Since $j_\mathcal{W}[\delta]$ is represented by $[\id]_\mathcal{W}$, the claim that for all $f : \mathcal{P}_\kappa(\lambda) \to \kappa$, $\{ z \in \mathcal{P}_\kappa(\delta) : f(z \cap \lambda) < \ot(z) \} \in \mathcal{W}$ is equivalent to the claim that $\sup\bar k[j_\mathcal{U}(\kappa)] \leq \delta$.
To show this, note:
\begin{align*}
\delta	&\geq		\sup\{ k(\alpha) : \alpha < j(\kappa) \} \\
		&\geq	\sup\{ k(\ell_\mathcal{U}(\alpha)) : \alpha < j_\mathcal{U}(\kappa) \} \\
		&=		\sup\{ \ell_\mathcal{W}(\bar k(\alpha)) : \alpha < j_\mathcal{U}(\kappa) \} \\
		&\geq	\sup\{ \bar k(\alpha) : \alpha < j_\mathcal{U}(\kappa) \}
\end{align*}

(\ref{amb-uf}) $\Rightarrow$ (\ref{amb}): Let $\lambda \geq \kappa$, let $\delta_0\geq\lambda$ be given by (\ref{amb-uf}), let $\delta\geq\delta_0$ and let $\mathcal{W}$ be a normal ultrafilter on $\mathcal{P}_\kappa(\delta)$ given by (\ref{amb-uf}).  Let $\pi : \mathcal{P}_\kappa(\delta) \to \mathcal{P}_\kappa(\lambda)$ be $z \mapsto z \cap \lambda$, and let $\mathcal{U}$ be the projection of $\mathcal{W}$ to $\mathcal{P}_\kappa(\lambda)$ via $\pi$.  Let $j : V \to M = \Ult(V,\mathcal{U})$ and $i : V \to N = \Ult(V,\mathcal{W})$ be the ultrapower embeddings, and let $k : M \to N$ be $[f]_\mathcal{U} \mapsto [f \circ \pi]_\mathcal{W}$.  Ordinals $\alpha\leq\lambda$ are represented in $\Ult(V,\mathcal{U})$ by the function $f_\alpha : z \mapsto \ot(z \cap \alpha)$, and $k(\alpha)$ is the $\mathcal{W}$-equivalence class of the function $z \mapsto \ot(z \cap \lambda \cap \alpha)$, which represents $\alpha$ in $\Ult(V,\mathcal{W})$.  Thus $\crit(k) > \lambda$.
For all $\alpha < j(\kappa)$, there is a function $f : \mathcal{P}_\kappa(\lambda) \to \kappa$ representing $\alpha$.  Since $\{ z \in \mathcal{P}_\kappa(\delta) : f\circ\pi(z) < \ot(z) \} \in \mathcal{W}$, $k(\alpha)<\delta$.

The implication (\ref{amb-uf}) $\Rightarrow$ (\ref{amb-uf-ptwise}) is trivial.  For the reverse direction, let $\lambda \geq \kappa$.  Let $\lambda' = 2^{\lambda^{<\kappa}}$, and let $\delta_0 \geq \lambda'$ witness (\ref{amb-uf-ptwise}) for $\lambda'$.  Enumerate all functions from $\mathcal{P}_\kappa(\lambda)$ to $\kappa$ as $\{ f_\alpha : \alpha < \lambda' \}$.  Let $g : \mathcal{P}_\kappa(\lambda') \to \kappa$ be $z \mapsto \sup\{ f_\alpha(z \cap \lambda) : \alpha \in z \}$.  Let $\delta \geq \delta_0$ and let $\mathcal{U}$ be a normal ultrafilter on $\mathcal{P}_\kappa(\delta)$ such that $\{ z \in \mathcal{P}_\kappa(\delta) : g(z \cap \lambda') < \ot(z) \} \in \mathcal{U}$.  By the fineness of $\mathcal{U}$, for all $\alpha<\lambda'$, $\{ z \in \mathcal{P}_\kappa(\delta) : f_\alpha(z \cap \lambda) < \ot(z) \} \in \mathcal{U}$.
\end{proof}

\begin{theorem}
\label{amb-ee}
Suppose $\kappa<\theta$ are inaccessible.  The following are equivalent:
\begin{enumerate}
\item\label{local-amb} $V_\theta \models \kappa$ is ambitious.
\item\label{mag-ee} For all structures $\frak A$ on $V_\theta$ in a countable language, there is a $\kappa$-Magidor $M \prec \frak A$ such that for all $\alpha<\kappa$, there is a $\kappa$-Magidor $N \prec \frak A$ such that $M \sqsubseteq N$ and $\ot(N \cap \theta) > \alpha$.
\end{enumerate}
Furthermore, in (\ref{mag-ee}), we can take $M$ and $N$ to be ${<}(M\cap\kappa)$-closed.
\end{theorem}

\begin{proof}
Suppose (\ref{local-amb}).  If (\ref{mag-ee}) fails, then there is a structure $\frak A$ on $V_\theta$ in a countable language extending $(V_\theta,\in,\kappa)$ such that for every $\kappa$-Magidor $M \prec \frak A$, there is a bound $b_{\frak A}(M) <\kappa$ on the order-type of $N \cap \theta$ whenever $N \prec \frak A$ is a $\kappa$-Magidor end-extension of $M$.

Let $\lambda<\theta$ be such that $V_\lambda \prec \frak A$ and $\cf(\lambda) \geq \kappa$.
Let $\delta_0 \geq \lambda$ witness ambitiousness.  Let $\delta \geq \delta_0$ be such that  $V_\delta \prec \frak A$ and $\cf(\delta) \geq \kappa$.
Let $\mathcal{W}$ be a normal ultrafilter on $\mathcal{P}_\kappa(V_\delta)$ such that for all $f : \mathcal{P}_\kappa(V_\lambda) \to \kappa$, $\{ z \in \mathcal{P}_\kappa(V_\delta) : f(z\cap V_\lambda) < \ot(z) \} \in \mathcal{W}$.  Let $\mathcal{U}$ be the projection of $\mathcal{W}$ to $\mathcal{P}_\kappa(V_\lambda)$, and let $j : V \to M$ be the ultrapower embedding via $\mathcal{U}$.  Then in $M$, $j[V_\lambda]$ is a $j(\kappa)$-Magidor elementary submodel of $j(V_\lambda) \prec j(\frak A)$.
It is ${<}\kappa$-closed by Lemma \ref{magclosure}.  Let $b = b_{\frak A} \restriction \mathcal{P}_\kappa(V_\lambda)$, and let $\xi = j(b)(j[V_\lambda])$.
Let $i : V \to N$ be the ultrapower map via $\mathcal{W}$, and let $k : M \to N$ be the factor map.  Then $\delta > k(\xi)$.
In $N$, $k(\xi) = i(b_{\frak A})(i[V_\lambda])$ is a bound on the order-types of $i(\kappa)$-Magidor end-extensions of $i[V_\lambda]$ that are elementary in $i(\frak A)$.  But $i[V_\delta]$ is an $i(\kappa)$-Magidor elementary submodel of $i(\frak A)$, $i[V_\delta] \sqsupseteq i[V_\lambda]$, and $\ot(i[V_\delta] \cap \theta) = \delta > k(\xi)$.  This contradiction shows that (\ref{mag-ee}) holds.

Now suppose (\ref{mag-ee}).  Suppose $\kappa \leq \lambda <\theta$, and let $\frak A$ be any structure in a countable language expanding $(V_\theta,\in,\kappa,\lambda)$.  Let $M \prec \frak A$ be a $\kappa$-Magidor model that can be end-extended to other $\kappa$-Magidor models elementary in $\frak A$ of arbitrarily high order-type below $\kappa$.  There is $\gamma < \kappa$, a structure $\bar{\frak A}$ on $V_\gamma$, and an elementary $j : \bar{\frak A} \to \frak A$, with $j[V_\gamma] = M$.  Let $\langle\bar\kappa,\bar\lambda\rangle = j^{-1}(\langle\kappa,\lambda\rangle)$.  Then we can define a normal ultrafilter $\mathcal{U}$ on $\mathcal{P}_{\bar\kappa}(\bar\lambda)$ by $X \in \mathcal{U}$ iff $j[\bar\lambda] \in j(X)$.  Of course, $\mathcal{U} \in V_\gamma$.  Let $j_\mathcal{U} : V_\gamma \to P = \Ult(V_\gamma,\mathcal{U})$ be the ultrapower embedding.  As usual, there is a factor map $\ell_0 : P \to V_\theta$ defined by $\ell_0([f]_{\mathcal{U}}) = j(f)(j[\bar\lambda])$, with $j = \ell_0 \circ j_\mathcal{U}$.  Let $\xi = \sup \ell_0[j_\mathcal{U}(\bar\kappa)]$, which is above $\bar\lambda$ since $j_\mathcal{U}(\bar\kappa) > \bar\lambda$.  Since $\gamma<\kappa$ and $\kappa$ is regular, $\xi <\kappa$.

Now let $N \prec \frak A$ be a $\kappa$-Magidor end-extension of $M$ with $\ot(N \cap \theta) > \xi$.  Let $\eta<\kappa$ be such that $N \cong V_\eta$.  Let $i : V_\eta \to V_\theta$ be the inverse of the transitive collapse map.  Then $i$ is an extension of $j$.  Let $\xi\leq\bar\delta<\eta$.  Let $\mathcal{W}$ be the normal ultrafilter on $\mathcal{P}_{\bar\kappa}(\bar\delta)$ defined by $X \in \mathcal{W}$ iff $i[\bar\delta] \in i(X)$.  Then $\mathcal{U}$ is the projection of $\mathcal{W}$ via the map $\pi : z \mapsto z \cap \bar\lambda$.  Let $j_{\mathcal{W}} : V_\eta \to Q = \Ult(V_\eta,\mathcal{W})$ be the ultrapower embedding, let $k : P \to Q$ be the factor map $[f]_\mathcal{U} \mapsto [f \circ \pi]_\mathcal{W}$, and let $\ell_1 : Q \to V_\theta$ be the map $[f]_\mathcal{W} \mapsto i(f)(i[\bar\delta])$.  We have the following commutative diagram:
\begin{diagram}
V_\gamma	&\rTo^{j_\mathcal{U}}	&P		&\rTo^{\ell_0}	&V_\theta	\\
\dTo^\id		&			&\dTo^k	&\ruTo^{\ell_1}	&		\\
V_\eta		&\rTo^{j_\mathcal{W}}	&Q		&			&
\end{diagram}
We have that:
\begin{align*}
\bar\delta	&\geq \sup\{ \ell_0(\alpha) : \alpha < j_\mathcal{U}(\bar\kappa) \} \\
		&=  \sup\{ \ell_1(k(\alpha)) : \alpha < j_\mathcal{U}(\bar\kappa) \} \\
		&\geq \sup\{ k(\alpha) : \alpha < j_\mathcal{U}(\bar\kappa) \}
\end{align*}
The relation $\bar\delta \geq \sup k[ j_\mathcal{U}(\bar\kappa) ]$ is equivalent to the statement that for all $f : \mathcal{P}_{\bar\kappa}(\bar\lambda) \to \bar\kappa$, $\{ z \in \mathcal{P}_{\bar\kappa}(\bar\delta) : f(z \cap \bar\lambda) < \ot(z) \} \in \mathcal{W}$, which can be computed in $V_\eta$.  Note that to arrange this property, we only needed to take $\bar\delta \geq \xi$.  By the elementarity of $i = \ell_1 \circ j_\mathcal{W}$,
for every $\delta \geq i(\xi)$ in $V_\theta$, there is a normal ultrafilter $\mathcal{W}'$ on $\mathcal{P}_\kappa(\delta)$
such that for all $f : \mathcal{P}_{\kappa}(\lambda) \to \kappa$, $\{ z \in \mathcal{P}_{\kappa}(\delta) : f(z \cap \lambda) < \ot(z) \} \in \mathcal{W}'$.
\end{proof}

In contrast to the class of countable models, Magidor models are more constrained in their end-extendibility.  Let us say a $\kappa$-Magidor model $M \prec V_\theta$ is \emph{$\theta$-maximal} when it cannot be end-extended to a larger $\kappa$-Magidor $N \prec V_\theta$.

\begin{prop}
Suppose $\kappa<\theta$ are inaccessible and $V_\theta \models \kappa$ is supercompact.  Then the set of $\theta$-maximal $\kappa$-Magidor models in $V_\theta$ is stationary.
\end{prop}

\begin{proof}
Suppose $\kappa$ is $\lambda$-supercompact.  Let $\mathcal{U},\mathcal{W}$ be two normal ultrafilters on $\mathcal{P}_\kappa(\lambda)$.  If $\mathcal{U} \in \Ult(V,\mathcal{W})$, then $j_\mathcal{U}(\kappa) < j_\mathcal{W}(\kappa)$ since $j_\mathcal{W}(\kappa)$ is inaccessible in $\Ult(V,\mathcal{W})$, and $\Ult(V,\mathcal{W})$ can compute $\Ult(\kappa,\mathcal{U})$.  Thus if $\mathcal{U}$ has $j_\mathcal{U}(\kappa)$ as small as possible, then $\kappa$ is not $\lambda$-supercompact in $\Ult(V,\mathcal{U})$.

Now let $\frak A$ be a structure on $V_\theta$ in a countable language.  Let $\lambda < \theta$ be such that $V_\lambda \prec V_\theta$.  Let $\mathcal{U}$ be a normal ultrafilter on $\mathcal{P}_\kappa(\lambda)$ with $j_\mathcal{U}(\kappa)$ as small as possible.  In $M = \Ult(V,\mathcal{U})$, $N_0 = j_\mathcal{U}[V_\lambda]$ is a $j_\mathcal{U}(\kappa)$-Magidor elementary submodel of $j(\frak A)$.  Working in $M$, if there were a $j_\mathcal{U}(\kappa)$-Magidor $N_1 \prec V^M_\theta$ end-extending $N_0$, then there would be an elementary $i : V_\eta^M \to V_\theta^M$, with $i[V_\eta^M] = N_1$ and $\eta > \lambda$.  Then we could define a normal ultrafilter $\mathcal{W}$ on $\mathcal{P}_\kappa(\lambda)$ by $X \in \mathcal{W}$ iff $i[\lambda] \in i(X)$.  This contradicts the observation of the previous paragraph that $\kappa$ is not $\lambda$-supercompact in $M$.  By the elementarity of $j_\mathcal{U}$, there is a $\theta$-maximal $\kappa$-Magidor $N \prec \frak A$.
\end{proof}

Note that the argument for the above proposition depended on finding a $\kappa$-Magidor model $M$ such that $M \cap \kappa$ and $\kappa$ differ on some large cardinal property; namely, $M \cap \kappa$ lacks a degree of supercompactness.
In \S\ref{limitations}, we will find other constraints on the end-extendibility of Magidor models that do not rely on such tricks.

To situate ambitiousness among the large cardinal notions, we show that it is strictly below almost-hugeness and strictly above the strongest notion found in the literature below almost-huge.  Perlmutter's paper \cite{perlmutter} seems to be the latest word on this.  A cardinal $\kappa$ is called \emph{high-jump} when for some $\lambda \geq \kappa$, there is a normal ultrafilter $\mathcal{U}$ on $\mathcal{P}_\kappa(\lambda)$ such that $\lambda\geq\sup\{ j_\mathcal{U}(f)(\kappa) : f : \kappa \to \kappa \}$.  Such an ultrafilter is called a \emph{high-jump measure}.    This notion was first introduced in \cite{srk} and given the name ``high-jump'' by \cite{ha}.  The strongest notion considered in \cite{perlmutter} below almost-huge is \emph{high-jump with unbounded excess closure}.  Perlmutter \cite[Proposition 4.6]{perlmutter} shows that this is equiconsistent with the existence of a cardinal $\kappa$ such that for all sufficiently large $\lambda \geq \kappa$, there is a high-jump measure on $\mathcal{P}_\kappa(\lambda)$.

\begin{prop}
Suppose $\kappa$ is almost-huge with target $\lambda$.  Then $V_\lambda \models$ ``$\kappa$ is ambitious,'' and $V_\kappa \models$ ``There are unboundedly many ambitious cardinals.''
\end{prop}

\begin{proof}
Let $j : V \to M$ be such that $\crit(j) = \kappa$, $j(\kappa) = \lambda$, and $M^{<\lambda} \subseteq M$.  Suppose $\kappa \leq \alpha < \lambda$.
Since $\lambda$ is inaccessible, there is $\delta_0 < \lambda$ such that for all $f : \mathcal{P}_\kappa(\alpha) \to \kappa$, $j(f)(j[\alpha])<\delta_0$.
Let $\delta_0\leq\delta<\lambda$ and let $\mathcal{U}$ be the normal ultrafilter on $\mathcal{P}_\kappa(\delta)$ derived from $j$.  Then for all $f : \mathcal{P}_\kappa(\alpha) \to \kappa$,
$\{ z \in \mathcal{P}_\kappa(\delta) : f(z \cap \alpha) < \ot(z) \} \in \mathcal{U}.$
The second claim follows from an easy reflection argument.
\end{proof}

\begin{prop}
If $\kappa$ is ambitious, then for all sufficiently large $\lambda \geq \kappa$, there is a high-jump measure on $\mathcal{P}_\kappa(\lambda)$.  Furthermore, the former property has strictly greater consistency strength than the latter.
\end{prop}

\begin{proof}
If $\kappa$ is ambitious, then there is $\delta_0 > \kappa$ such that for all $\delta\geq\delta_0$, there is a normal ultrafilter $\mathcal{U}$ on $\mathcal{P}_\kappa(\delta)$ such that for all $f : \kappa \to \kappa$, $\{ z \in \mathcal{P}_\kappa(\delta) : f(z \cap \kappa) < \ot(z) \} \in \mathcal{U}$.  This means that if $j : V \to M$ is the ultrapower embedding via $\mathcal{U}$, then for all $f : \kappa \to \kappa$, $j(f)(\kappa) < \delta$.  Thus $\mathcal{U}$ is a high-jump measure.

To show the strict consistency comparison, we establish that there are unboundedly many inaccessible cardinals above an ambitious cardinal $\kappa$.  Then if $\theta$ is any inaccessible above the witness $\delta_0$ for $\kappa$, then $V_\theta \models$ ZFC +  ``For all $\delta\geq\delta_0$, there is a high-jump measure on $\mathcal{P}_\kappa(\delta)$.''  To this end, let $\lambda \geq \kappa$ be arbitrary.  Let $j : V \to M$, $k : M \to N$ be as in the definition of ambitiousness.  There is $\theta$ such that $\lambda < \theta < j(\kappa)$ and $M \models \theta$ is inaccessible.  There is $\delta$ such that $N$ is $\delta$-closed and $k(\theta) < \delta$.  Thus $k(\theta)$ is inaccessible in $V$, and $k(\theta) > \lambda$.
\end{proof}

\section{Neeman forcing with Magidor models}
\label{sec:wcctp}

This section contains our main results.  We use Neeman forcing with Magidor models as the small type to force the weak Chang's Conjecture along with the tree property and derive some limitative results about Magidor models.

Suppose $\kappa<\theta$ are inaccessible and $\mathbb Q \subseteq V_\kappa$ is $\kappa$-c.c.  Let $\frak A$ be a structure on $V_\theta$ in a countable language with a constants for $\mathbb Q$ and $\kappa$.  Then for every $M \prec \frak A$ with $M \cap \kappa \leq \kappa$ and $\mathbb Q \in M$, $1_{\mathbb Q}$ is a master condition for $M$.  Suppose that $\mathcal{S}_0$ is a set of $\kappa$-Magidor elementary submodels of $\frak A$.  Let $\mathcal{T}$ be the set of all $V_\alpha \prec \frak A$ such that $\cf(\alpha) \geq \kappa$.   Let $\mathcal{S}$ be the set of all models of the form $M \cap V_\alpha$ for $M \in \mathcal{S}_0$ and $V_\alpha \in \mathcal{T}$.  Note that by Lemma \ref{rankmeet}, each $M \in \mathcal{S}$ is elementary in $\frak A$, and since it is an initial segment of an $N \in \mathcal{S}_0$, it is also $\kappa$-Magidor.

Now let $H \subseteq \mathbb Q$ be generic over $V$.  For a set of $\mathcal{R} \in V$ of elementary submodels of $\frak A$, define in $V[H]$, $\mathcal{R}[H] = \{ M[H] : M \in \mathcal{R} \}$.  By the properness of $\mathbb Q$ for $\mathcal{S}\cup\mathcal{T}$, for each $M \in \mathcal{S} \cup \mathcal{T}$, $M[H] \cap V = M$.  Furthermore, $\mathcal{T}[H]$ satisfies the definition of $\mathcal{T}$ in $V[H]$, i.e.\ $W \in V[H]$ is a rank below $V_\theta$ elementary in $\frak A[H]$ with $\cf(W \cap \theta) \geq \kappa$ iff $W = V_\alpha[H]$ for some $V_\alpha \prec \frak A$ with $\cf(\alpha) \geq \kappa$.  It is also easy to see that $\mathcal{S}_0$ is stationary in $V$ iff $\mathcal{S}_0[H]$ is stationary in $V[H]$.
If $\mu<\kappa$ is regular in both $V$ and $V[H]$ and $\mu \in M$ for every $M \in \mathcal{S}$, then $\langle \mu,\kappa,\theta,\langle V_\alpha[H] : \alpha<\theta\rangle,\mathcal{S}[H],\mathcal{T}[H] \rangle$ is ``nice'' per the definition in \S\ref{sec:neeman}.

It will simplify things to replace $\mathbb P^{\mathrm{dec}*}_{\mu,\mathcal{S}[H],\mathcal{T}[H]}$ with an isomorphic copy $\mathbb P$ whose elements are closer to $V$.  Let us assume that $\frak A$ includes a well-order.  Let us take $\mathbb P$ to be the set of $\langle s,f \rangle$ such that:
\begin{enumerate}
\item $s$ is a ${<}\mu$-sized subset of $\mathcal{S} \cup \mathcal{T}$ that is linearly ordered by rank and closed under intersections.
\item If $\langle M_i : i < \alpha \rangle$ orders $s$ by rank, then for each $\beta<\alpha$,  $\{ i < \beta : M_i \in M_\beta \}$ is cofinal in $\beta$.
\item\label{hs1} For each $M \in s$, $s \cap M \in M[H]$.
\item $f$ is a function with a countable domain contained $s \times \mu$ such that:
\begin{enumerate}
\item If $\langle M,\alpha \rangle \in \dom(f)$, $f(M,\alpha) \in M^* \cap W \cap \theta$, where $M^*$ is the successor of $M$ in $s$ if it exists, $M^* = V_\theta$ if $M$ is the largest model in $s$, and $W$ is the smallest member of $\mathcal{T}$ such that $M \in W$.
\item\label{hs2} For each $M \in s$, $f \cap M \in M[H]$.
\end{enumerate}
\end{enumerate}
Thus conditions in $\mathbb P$ are pairs of ${<}\mu$-sized subsets of $V$ satisfying certain conditions.
The reason that we can require the decorations to be functions with range in the ordinals, and get something isomorphic to our original poset, is because of the well-ordering in the background.  The ground-model well-order $\lhd$ has a canonical extension to a well-order $\lhd'$ of $V_\theta[H]$ by letting $x \lhd' y$ when $\tau$ is the $\lhd$-least name such that $x = \tau^H$, $\sigma$ is the $\lhd$-least name such that $y = \sigma^H$, and $\tau \lhd \sigma$.  So for $M[H] \prec \frak A[H]$, there is a canonical translation between functions into $M[H]$ and functions into $M \cap \theta$.

\subsection{A forcing characterization of ambitiousness}

Suppose $\kappa<\theta$ are inaccessible, $\mu<\kappa$ is regular, $\mathbb Q \subseteq V_\kappa$ is $\kappa$-c.c., and $\mathbb Q$ preserves the regularity of $\mu$.
Let $\frak A$ be a structure on $V_\theta$ in a countable language that has constants for $\mu$ and $\mathbb Q$ and includes a well-order,
let $\mathcal{S}$ be the set of \emph{all} $\kappa$-Magidor $M \prec \frak A$ that are ${<}(M \cap \kappa)$-closed, and let $\mathcal{T}$ be the set of all $V_\alpha \prec \frak A$ with $\cf(\alpha)\geq\kappa$.  If $H \subseteq \mathbb Q$ is generic, define $\mathbb M^{\frak A}(\mu,\kappa,\theta)$ to be $\mathbb P^{\mathrm{dec}*}_{\mu,\mathcal{S}[H],\mathcal{T}[H]}$.
With these assumptions and notations fixed, we have:

\begin{theorem}
\label{forcewcc}
Suppose that $\mathbb Q * \dot{\mathbb M}^{\frak A}(\mu,\kappa,\theta)$ preserves the class $\cof({\geq}\mu)$.
Then $\mathbb Q * \dot{\mathbb M}^{\frak A}(\mu,\kappa,\theta)$ forces $\wcc(\mu^+,\cof(\mu))$ if and only if $V_\theta \models$ ``$\kappa$ is ambitious.''
\end{theorem}

\begin{proof}
Let us work with the isomorphic copy $\mathbb P$ of $\mathbb M^{\frak A}(\mu,\kappa,\theta)$ described above.

First suppose $V_\theta \models$ ``$\kappa$ is ambitious.''  By Lemma \ref{cards},  $\mathbb Q * \dot{\mathbb P}$
forces that  $\kappa = \mu^+$ and $\theta = \mu^{++}$.  For every $\kappa$-Magidor $M \prec V_\theta$, $V \models \mu < \cf(M \cap \kappa) < \kappa$, so $\mathbb Q * \dot{\mathbb P}$
forces that $\cf(M \cap \kappa) = \mu$.  By the size of $\mathbb Q$, may assume that $\mathbb Q * \dot{\mathbb P}
\subseteq V_\theta$.  Now let $\dot F$ be a $\mathbb Q * \dot{\mathbb P}$
-name for a function from $\theta^{<\omega}$ to $\theta$, let $\dot f$ be a name for a function from $\kappa$ to $\kappa$, and let $\langle q_0,\dot p_0 \rangle \in \mathbb Q * \dot{\mathbb P}$
be arbitrary.  Let $\frak B = \langle H_{\theta^+},\in,\frak A,\dot F,\dot f,\langle q_0,\dot p_0\rangle,\lhd \rangle$, where $\lhd$ is a well-order.  Let $\frak C$ be the result of restricting the Skolem functions for $\frak B$ to $V_\theta$.  Let $F' : V_\theta^{<\omega} \to V_\theta$ be another function.  By Theorem \ref{amb-ee}, there is a $\kappa$-Magidor $M \prec \frak C$ that is ${<}(M\cap\kappa)$-closed, $M$ is closed under $F'$, and for every $\alpha<\kappa$, there is a $\kappa$-Magidor $N \prec \frak C$ such that
$M \sqsubseteq N$, $N$ is ${<}(N\cap\kappa)$-closed, and $\ot(N \cap \theta)>\alpha$.

It follows that there are stationary-many $\kappa$-Magidor $M \prec \frak C$ that are ${<}(M\cap\kappa)$-closed and can be end-extended to $\kappa$-Magidor $N \prec \frak C$ with the same closure such that $\ot(N \cap \theta)$ is arbitrarily large below $\kappa$.  Let $\ensuremath{\mathcal{E}}$ be the set of such $M$. For each $\alpha<\theta$ choose $M_\alpha \in \ensuremath{\mathcal{E}}$ such that $\alpha \in M_\alpha$.  If $\cf(\alpha) \geq \kappa$, then $\sup(M_\alpha \cap \alpha) < \alpha$.  By Fodor's Lemma and the inaccessibility of $\theta$, there is a stationary $X \subseteq \theta \cap \cof({\geq}\kappa)$ and an $M^*$ such that for every $\alpha \in X$, $M_\alpha \cap \alpha = M^*$.  We may assume that for every $\alpha \in X$, $V_\alpha \prec \frak C$.  It follows that $M^* \prec \frak C$, $M^*$ is $\kappa$-Magidor, and $M^*$ is ${<}(M^*\cap\kappa)$-closed.

It is forced that $\dot p_0$ can be extended to include the model $M^*$.  Let $\dot p_1$ be a $\mathbb Q$-name for such an extension.  Let $\langle q_1,\dot p_2 \rangle \leq \langle q_0,\dot p_1 \rangle$ decide the value of $\dot f(M^*\cap\kappa)$, say as $\xi<\kappa$.  Let $\alpha \in X$ be such that $\langle q_1,\dot p_2 \rangle \in V_\alpha$.  Let $N \sqsupseteq M_\alpha$ be such that $N$ is $\kappa$-Magidor, $N \prec \frak C$, $N$ is ${<}(N\cap\kappa)$-closed, and $\ot(N \cap \theta)> \xi$.  Let $\dot s_2$ be a name for the set of models appearing in $\dot p_2$.  Let $\dot s_3$ be a name such that $\Vdash_{\mathbb Q} \dot s_3 = \dot s_2 \cup \{ V_\alpha,N \}$.  Then $\dot s_3$ is forced to be closed under intersections since $N \cap V_\alpha = M^*$, and it is forced that if $R \in \dot s_2$, then $V_\alpha \cap R = R$, and $N \cap R = N \cap (V_\alpha \cap R) = M^* \cap Q \in \dot s_2$.
Furthermore, it is forced that $\dot s_3 \cap N = (\dot s_3 \cap M^*) \cup \{ V_\alpha \}$, which is forced to be a member of $N$.
If $\dot h$ is a name for the decoration of $\dot p_2$, let $\dot p_3$ be a name for $\langle \dot s_3,\dot h \rangle$.  Then $\langle q_1,\dot p_3 \rangle \leq \langle q_1,\dot p_2 \rangle$, and $\langle q_1,\dot p_3 \rangle$ is a master condition for $\sk^{\frak B}(N)$.  Thus $\langle q_1,\dot p_3 \rangle$ forces that $N$ is closed under $\dot F$ and that $\dot f(N \cap \kappa) < \ot(N \cap \theta)$.  By the arbitrariness of $\dot F$, $\dot f$, and $\langle q_0,\dot p_0\rangle$, $\wcc(\kappa,\cof(\mu))$ is forced.

Now assume that $V_\theta \models$ ``$\kappa$ is not ambitious.''  We have two cases:

\textbf{Case 1}:  $V_\theta \models$ ``$\kappa$ is not supercompact.''  Then there is $\lambda_0 \in [\kappa,\theta)$ such that for all $\lambda \geq \lambda_0$, there is no normal fine ultrafilter on $\mathcal{P}_\kappa(\lambda)$.  It follows that for every $\lambda \in [\lambda_0,\theta)$, there is no $\kappa$-Magidor $M \prec V_\theta$ with $\lambda \in M$.  This is because if there were such an $M$, then there would be an elemetnary $i : V_\eta \to V_\theta$ for some $\eta<\kappa$, with $i(\crit(i)) = \kappa$.  If $i(\bar\kappa) = \kappa$ and $i(\bar\lambda) = \lambda$, then we could define a normal fine ultrafilter $\mathcal{U}$ on $\mathcal{P}_{\bar\kappa}(\bar\lambda)$ by $X \in \mathcal{U}$ iff $i[\bar\lambda] \in i(X)$.  But then $\mathcal{U} \in V_\eta$, and by elementarity, $V_\theta \models$ ``$\kappa$ is $\lambda$-supercompact,'' a contradiction.

Let $H \subseteq \mathbb Q$ be generic.  In $V[H]$, for any $p \in\dot{\mathbb P}$
the only models included in $p$ of rank $\geq \lambda_0$ are transitive models.  Thus if $V_\alpha \in V_\beta$ are two consecutive transitive models, with $\alpha \geq \lambda_0$, and $G \subseteq \mathbb P$
is generic over $V[H]$, then the union of the decorations attached to $V_\alpha$ that appear in $G$ constitutes a surjection from $\mu$ to $V_\beta$.  But notice also that Corollary \ref{thetacc} still holds for $\dot{\mathbb P}$,
and so $\theta = \mu^+$ in $V[H][G]$.  Now in $V$, let $f : \theta \to \theta$ be the function $\beta \mapsto \beta^+$.  Let $\alpha<(\theta^+)^V = (\theta^+)^{V[H][G]}$ and let $\sigma : \theta \to \alpha$ be a sujrection in $V$.  For all $\beta < \theta$, $\ot(\sigma[\beta]) <  (\beta^+)^V = f(\beta)$.  Thus in $V[H][G]$, $f$ dominates each canonical function pointwise, and so $\wcc(\theta,A)$ fails for every $A \subseteq \theta$.

\textbf{Case 2}:  $V_\theta \models$ ``$\kappa$ is supercompact but not ambitious.''  In this case, the set $\mathcal{S}$ is stationary,
so $\mathbb Q * \dot{\mathbb P}$
forces that $\kappa = \mu^+$ and $\theta = \mu^{++}$.  By Theorem \ref{amb-ee}, there is a structure $\frak B$ on $V_\theta$ in a countable language and a function $b : V_\theta \to \kappa$ such that for all $\kappa$-Magidor $M \prec \frak B$, if $N \prec \frak B$ is $\kappa$-Magidor and $M \sqsubseteq N$, then $\ot(N \cap \theta) < b(M)$.  We may assume that $\frak B$ has definable Skolem functions.

Let $H*G \subseteq \mathbb Q * \dot{\mathbb P}$
be generic.  $G$ yields a chain of models $\langle M_\alpha : \alpha < \theta \rangle$.  For $\alpha < \kappa$ such that $\alpha = M \cap \kappa$ for some model $M \prec \frak B$ appearing in $G$, let $M_\alpha$ be the first such model in the chain.  We claim that for all $N \prec \frak B$ appearing in $G$ with $N \cap \kappa = \alpha$, $M_\alpha \sqsubseteq N$.  If $N$ is such a model, then $M_\alpha \cap N$ also appears in $G$, and since $M_\alpha \cap N \prec \frak B$ (because it is closed under the Skolem functions for $\frak B$) and $M_\alpha \cap N \cap \kappa = \alpha$, we must have that $M_\alpha \cap N = M_\alpha$ by the minimality of $M_\alpha$.
If $N \not= M_\alpha$, then $M_\alpha \notin N$, so there must be transitive models appearing between them.
Let $W$ be the least such transitive model.  Then $W \cap N$ also appears in $G$.  If $W \cap N \not= M_\alpha$ then $W \cap N$ occurs either before or after $M_\alpha$.  If were to occur after, then since there are no transitive models between $M_\alpha$ and $W \cap N$, we would have $M_\alpha \in W \cap N$.  But this is impossible, since $M_\alpha \cap \kappa = W \cap N \cap \kappa$.
But also $W \cap N$ cannot occur before $M_\alpha$, since $M_\alpha \cap N = M_\alpha$, so $M_\alpha \subseteq W \cap N$ and thus $\rank(M_\alpha) \leq \rank(W \cap N)$.  Thus $M_\alpha = W \cap N  \sqsubseteq N$.

In $V[H][G]$, define $f : \kappa \to \kappa$ by $f(\alpha) = b(M_\alpha)$ if $M_\alpha$ is defined, and otherwise $f(\alpha) = 0$.  Now let $X \subseteq \kappa \cap \cof(\mu)$ be a stationary set in $V[H][G]$.  Since $\mathbb Q$ preserves that $\cf(M \cap \kappa) \geq \mu$ for all $M \in \mathcal{S}$, Lemma \ref{*proj} implies that the set of $M$ appearing in $G$ with $M \cap \kappa \in X$ is stationary.  Let $\gamma < \theta$, and let $\sigma : \kappa \to \gamma$ be a surjection in $V[H][G]$.  Let $M^* \prec H_\theta^{V[H][G]}$ be such that $M^* \cap V = M \prec \frak B$, $M$ appears in $G$, $M \cap \kappa = \alpha \in X$, and $\{ f,\sigma \} \in M^*$.  Then $\ot(M \cap \theta) < b(M_\alpha) = f(\alpha)$.  If $\psi_\gamma$ is the canonical function for $\gamma$ defined using $\sigma$, then $\psi_\gamma(\alpha) = \ot(M\cap\gamma)< \ot(M \cap \theta) < f(\alpha)$.  Since $X$ was an arbitrary stationary subset of $\kappa \cap \cof(\mu)$, the set of $\beta \in \kappa \cap \cof(\mu)$ such that $\psi_\gamma(\beta) \geq f(\beta)$ is nonstationary.  Thus $f$ dominates all canonical functions modulo clubs on $\kappa \cap \cof(\mu)$, so $\wcc(\kappa,\cof(\mu))$ fails.
\end{proof}

\subsection{The tree property}
\label{sec:tp}

In \cite[\S 5.1]{neeman}, Neeman gave a very brief description of an unpublished result of his that if $\kappa$ is supercompact and $\theta>\kappa$ is weakly compact, then the tree property can be forced simultaneously at $\omega_2$ and $\omega_3$ by a two-step iteration of the form $\mathbb P * \dot{\mathbb Q}$, where $\mathbb P$ is the finite-conditions two-types poset making $\kappa = \omega_2$, and $\dot{\mathbb Q}$ is the countable-conditions two-types poset with former $\kappa$-Magidor models.  The author is grateful to Neeman for patiently explaining his argument.  We present a proof here for a slightly simpler situation where we replace the first step with $\add(\omega,\kappa)$ and use our modified decorated poset in the second step.

The main difficulty is proving that the tree property at $\kappa$ holds after this forcing.  For this conclusion, it is not necessary for $\theta$ to be weakly compact.  But if it is, the tree property at $\theta$ will hold by the argument of \cite[\S 5.1]{neeman}.

\begin{theorem}
\label{tpthm}
Suppose that $\theta$ is inaccessible, $\kappa<\theta$ is $\theta$-supercompact, and $\frak A = \langle V_\theta,\in,\kappa,\lhd \rangle$, where $\lhd$ is a well-order.
Then $\add(\omega,\kappa) * \dot{\mathbb M}^{\frak A}(\omega_1,\kappa,\theta)$ forces that $2^\omega = \omega_2 = \kappa$ and the tree property holds at $\kappa$.
\end{theorem}

Before beginning the argument, we make the following remark.  We will show that all the reals added in the two-step iteration are added in the first step.  One might expect that $\add(\omega,\kappa)$ could be replaced by a general type of poset, say one that is proper, $\kappa$-c.c., and contained in $V_\kappa$, and obtain the same conclusion.  For example, we might consider replacing the first step with a standard forcing for the proper forcing axiom (PFA) such as Baumgartner's (see \cite{jech}).  However, a well-known result of Todor\v{c}evi\'{c} \cite{todo} says that under PFA, every forcing that adds a subset of $\omega_1$ either adds reals or collapses the continuum.  If $\mathbb B$ is Baumgartner's forcing and $H \subseteq \mathbb B$ is generic, then the forcing $\mathbb M^{\frak A}(\omega_1,\kappa,\theta)$
adds generic subsets of $\omega_1$ over $V[H]$, and since it preserves $\kappa$, it must add reals over $V[H]$.  We do not know whether $\omega_1$ is preserved in this scenario.  The key property enabling distributivity when the first step is $\add(\omega,\kappa)$ seems to be having many strongly proper quotients, which plays a crucial role in Lemma \ref{magic} below.

\begin{proof}[Proof of Theorem \ref{tpthm}]
Let us work with the isomorphic copy $\mathbb P$ of $\mathbb M^{\frak A}(\omega_1,\kappa,\theta)$ described at the beginning of the section.  Let $\mathcal{S}$ be the set of all $\kappa$-Magidor $M \prec \frak A$ and let $\mathcal{T}$ be the set of all $V_\alpha\prec\frak A$ with $\cf(\alpha)\geq\kappa$.

\begin{claim}
\label{intermediateclosure}
For any $M \prec \frak A$, and any countable $x \in V[H]$ contained in $M$, if $x \in M[H]$, then $x \in V[H \cap M]$.  Furthermore, if $M$ is countably closed in $V$, then $x \in M[H]$ if and only if $x \in V[H \cap M]$.
\end{claim}

\begin{proof}
Let $\pi : M[H] \to \bar M[\bar H]$ be the transitive collapse.  Note that $\pi \restriction M$ is definable in $V$.
Suppose first that $x \in M[H]$.  By the definition of the transitive collapse, $\pi(x)$ is the pointwise image $\pi[x]$.
Since $x$ can be computed from $\pi \restriction M$ and $\pi(x) \in \bar M[\bar H] \subseteq V[\bar H]$, it follows that $x \in V[\bar H] = V[H \cap M]$.

Now suppose that $M$ is countably closed in $V$ and $x \in V[H \cap M]$ is a countable subset of $M$.  Then $\bar M$ is countably closed, and by the chain condition, $\bar M[\bar H]$ is countably closed in $V[\bar H] = V[H \cap M]$.   Since $\pi[x]$ is a countable subset of $\bar M$ that lives in $V[\bar H]$, it is in $\bar M[\bar H]$.  It follows that $x = \pi^{-1}(\pi[x]) \in M[H]$.
\end{proof}

\begin{claim}
\label{posetmeetmodel}
For any $M \prec \frak A$, $\mathbb P \cap M[H]$ belongs to $V[H \cap M]$.
Furthermore, if $\theta^* > \theta$ is regular and $\frak A \in M \prec H_{\theta^*}$, then for any $X \subseteq \mathbb P$ in $M[H]$, $X \cap M[H] \in V[H \cap M]$.
\end{claim}

\begin{proof}
By Claim \ref{intermediateclosure}, each $p \in \mathbb P \cap M[H]$ is a member of $V[H \cap M]$.  We must see that the conditions for membership in $\mathbb P$ can be expressed in $V[H \cap M]$. The clauses other than (\ref{hs1}) and (\ref{hs2}) are easily definable, so suppose $\langle s,f \rangle \in M[H]$ is a pair satisfying all of the clauses except perhaps these two.  Let $\pi : M[H] \to \bar M[\bar H]$ be the transitive collapse.  Then $\langle s,f \rangle \in \mathbb P$ if and only if for all $N \in s$, $s \cap N \in N[H]$ and $f \cap N \in N[H]$, which happens if and only if $\pi(s) \cap \pi(N) \in \pi(N)[\bar H]$ and $\pi(f) \cap \pi(N) \in \pi(N)[\bar H]$.  The latter condition can be defined in $V[\bar H] = V[H \cap M]$, so $\mathbb P \cap M[H]$ can be defined in $V[H \cap M]$.

For the second claim, suppose $\tau \in M$ is a name and $X = \tau^H$ is a subset of $\mathbb P$.  If $p \in M[H] \cap X$, then we can take a countable name $\sigma \in M$ such that $\sigma^H = p$, and thus in fact $\sigma \subseteq M$ and $\sigma^H = \sigma^{H \cap M}$.
Since $H$ is generic over $M$, some $q \in H \cap M$ forces $\sigma \in \tau$.  Thus $X \cap M[H]$ can be defined in $V[H \cap M]$ as the set of $p \in \mathbb P \cap M[H]$ such that for some countable $\sigma \in M$ satisfying $\sigma^{H \cap M} = p$, some $q \in H \cap M$ forces $\sigma \in \tau$.
\end{proof}

Both countable distributivity and the tree property will follow from the ability to build complete master conditions for a stationary class of countable models.  The following lemma enables this by granting us some ``free genericity.''  In $V$, let $\theta^* > \theta$ be regular, and let $N \prec H_{\theta^*}$ be countable with $\frak A \in N$.

\begin{lemma}
\label{magic}
Suppose $M \in \mathcal{S} \cap N$ and $G$ is a filter over $\mathbb P \cap M[H] \cap N[H]$ that belongs to $V[H \cap M]$ and meets every dense subset of $\mathbb P \cap M[H]$ that belongs to $N[H\cap M]$.  Then $G$ meets every dense subset of $\mathbb P \cap M[H]$ that belongs to $N[H]$.
\end{lemma}

\begin{proof}
Let $D \in N[H]$ be a dense subset of $\mathbb P \cap M[H]$.  By Claim \ref{posetmeetmodel}, $D \cap N[H]$ belongs to $V[H \cap N]$.
The objects $G$ and $D$ are in the model $V[H \cap M][H \cap N]$, which is a forcing extension of $V[H \cap M]$.  Let $\mathbb R$ be the quotient forcing $\add(\omega,\kappa \setminus M)$.  Let $\tau \in N[H \cap M]$ be an $\mathbb R$-name for $D$, and let $q \in H \cap N$ decide whether $G \cap \tau = \emptyset$.
In $V[H \cap M]$, define:
$$E = \{ p \in \mathbb P \cap M[H] : (\exists r \leq q)r \Vdash p \in \tau \}$$
This set is dense, and it is definable in $N[H \cap M]$.  It follows that $G$ meets $E$, say at $p$.  Thus some $r \leq q$ forces $p \in \tau$, and so $q$ must already force that $G \cap \tau \not= \emptyset$.
\end{proof}

\begin{lemma}
\label{genericsexist}
Suppose $M \in \mathcal{S} \cap N$ and $G\subseteq N[H]$ satisfy:
\begin{enumerate}
    \item[(a)] $G$ is a $(\mathbb P \cap M[H])$-generic filter over $N[H]$;
    \item[(b)] if $G = \{ \langle s_n,f_n \rangle : n \in \omega \}$ then $m_G := \langle \bigcup_n s_n, \bigcup_n f_n \rangle$ is a condition in $\mathbb P\cap M[H]$.
\end{enumerate}
Suppose $M^* \in \mathcal{S}$, $\rank(M)<\rank(M^*)$, and $p\in\mathbb P \cap N[H]$ is a condition such that $M$ and $M^*$ appear in $p$ and $p \restriction M \in G$.
Then there exists $G^*$ such that:
\begin{enumerate}
\item $p \restriction M^* \in G^*$;
\item $G^* \cap (M \cap M^*)[H] = G \cap (M \cap M^*)[H]$;
    \item $G^*$ is a $(\mathbb P \cap M^*[H])$-generic filter over $N[H]$;
    \item if $G^* = \{ \langle s^*_n,f^*_n \rangle : n \in \omega \}$ then $m_{G^*} := \langle \bigcup_n s^*_n, \bigcup_n f^*_n \rangle$ is a condition in $\mathbb P \cap M^*[H]$.
\end{enumerate}
\end{lemma}

\begin{proof}
We argue by induction on the rank of the models $M$ and $M^*$.  Let $M,M^*,G,p$ be given, and suppose that the claim holds for configurations where each model has rank less than $\rank(M^*)$.

Let us first note that $m_G \restriction (M \cap M^*)[H] \in (M \cap M^*)[H]$.  If $M \cap \kappa \leq M^*\cap\kappa$, then this follows from Claim \ref{intermediateclosure}, since:
$$m_G \in M[H]	\Rightarrow m_G \in V[H \cap M] \Rightarrow m_G \restriction (M \cap M^*) \in V[H \cap M]$$
$$\Rightarrow m_G \restriction (M \cap M^*) \in (M \cap M^*)[H]$$
On the other hand, if $M\cap\kappa > M^*\cap \kappa$, then $M \cap M^* \in M$ by Lemma \ref{bobanclaim}.  Then $m_G$ is a condition in which $M \cap M^*$ appears, so $m_G \restriction (M \cap M^*) \in (M\cap M^*)[H]$.

Now suppose that $M^*$ satisfies the following:

\begin{itemize}
\item[\Coffeecup]
For every $q \leq p \restriction M^*$ in $M^*[H]$, there is $R \in \mathcal{S} \cap M^*$ such that $q \in R[H]$.
\end{itemize}
In this case, we can proceed as follows.  Working in $V[H \cap M^*]$,
enumerate the dense subsets of $\mathbb P \cap M^*[H]$ that belong to $N[H \cap M^*]$ as $\langle D_i : 1\leq i < \omega \rangle$.
Let $G_0 = G \cap (M \cap M^*)[H]$, and let $p_0 = p \restriction M^*$.
$\{ d \restriction (M \cap M^*) : d \in D_1$ and $d \leq p_0 \}$ is dense in $\mathbb P \cap (M \cap M^*)[H]$ below $p_0 \restriction (M \cap M^*)$.
Thus we can find $p_1 \in D_1$ such that $p_1 \leq p_0$ and $p_1 \restriction (M \cap M^*) \in G_0$.  Then find $R_1 \in \mathcal{S} \cap M^*$ such that $p_1 \in R_1[H]$.  Using the induction hypothesis, let $G_1$ be a filter over $\mathbb P \cap R_1[H] \cap N[H]$ that possesses $p_1$, meets all dense sets in $N[H \cap M^*]$, such that $m_{G_1}$ is a condition in $R_1[H]$, and $G_1 \cap (M \cap M^*)[H] = G_0$.  Then take $p_2 \leq p_1$ such that $p_2 \in D_2$, $R_1$ appears in $p_2$, and $p_2 \restriction R_1 \in G_1$.
Continue in this way, producing a sequence of models $R_i \in \mathcal{S} \cap N$, a descending sequence of conditions $p_i$, and a sequence of filters $G_i$,
such that for each $i \geq 1$,
$p_i \in D_i$,
$p_i \in R_i[H] \cap G_i$,
$R_i$ appears in $p_{i+1}$,
$G_i$ is $(N[H \cap M^*],\mathbb P \cap R_i[H])$-generic,
and $m_{G_i}$ is a condition in $R_i[H]$.

Let $G^*$ be the filter generated by the sequence of $p_i$'s.
If $p_i = \langle s_i,f_i \rangle$, define $p_\omega = \langle s_\omega,f_\omega \rangle$ as $\langle \bigcup_i s_i,\bigcup_i f_i \rangle$.
Since the sequence of $p_i$'s was constructed in $V[H \cap M^*]$, $p_\omega$ is a member of $M^*[H]$ by Claim \ref{intermediateclosure}.
We must check that it is a condition.
For any model $R \in \mathcal{S}$ that appears in $p_\omega$, there is $i$ such that $R \in R_i$, so it suffices to show that $p_\omega \restriction R_i = m_{G_i}$, since then $p_\omega \restriction R = (p_\omega \restriction R_i) \restriction R = m_{G_i} \restriction R$, which is a condition since $R$ appears in $m_{G_i}$.
Let $m_{G_i} = \langle s_{G_i},f_{G_i} \rangle$.
For each $n < \omega$, $p_n \restriction R_i \in G_i$, so $s_\omega \cap R_i \subseteq s_{G_i}$ and $f_\omega \cap R_i \subseteq f_{G_i}$.
Furthermore, since the filter generated by the sequence $\langle p_n \restriction R_i : n < \omega \rangle$ is sufficiently generic, it must be maximal,
so $s_\omega \cap R_i = s_{G_i}$ and $f_\omega \cap R_i = f_{G_i}$.
By construction, the filter $G^*$ generated by $p_\omega$ meets all dense sets in $N[H \cap M^*]$. Lemma \ref{magic} allows us to conclude that $G^*$ is fully generic over $N[H]$.

Let us now consider cases depending on the position of $M^*$.  Let $C \subseteq \theta$ be the closure of $\{ \alpha : V_\alpha \in \mathcal{T} \}$.

\textbf{Case 1:} $C$ is cofinal in $M^* \cap \theta$.

In this case, for every $q \leq p \restriction M^*$ in $M^*[H]$, there is $V_\alpha \in \mathcal{T} \cap M^*$ such that $q \in V_\alpha[H]$.  Since $\mathcal{S} \cap V_\alpha$ is $\subseteq$-cofinal in each $V_\alpha \in \mathcal{T}$, and $M^*$ knows this by elementarity, \Coffeecup\, applies.

\textbf{Case 2:} There is a largest element $\gamma \in C$ below $\rank(M^*)$.

\textbf{Case 2.1:} For every $q \leq p \restriction M^*$ in $M^*[H]$, there is $q' \leq q$ in $M^*[H]$ such that $q'$ has a model $R \in \mathcal{S} \cap M^*$ of rank greater than all models appearing in $q$.

In this case, let us take such $q$ that has a largest model $R$ above the maximum point of $C$ below $\rank(M^*)$.  Then for any $x \in M^*$, we can extend $q$ by adding $x$ to the decoration on $R$, and by hypothesis, there is some stronger condition $q'$ with a model $R'$ above $R$.  Thus $x \in R'$, and \Coffeecup\, applies.

\textbf{Case 2.2:} There is $q \leq p \restriction M^*$ in $M^*[H]$ that has a largest-rank model $R$, and $q$ cannot be extended to a stronger condition in $M^*[H]$ with a model above $R$.

\textbf{Case 2.2.1:} $V_\gamma \in \mathcal{T}$.

Work in $V[H \cap M^*]$.  First use the induction hypothesis to obtain a filter $G_1 \subseteq \mathbb P \cap (M^* \cap V_\gamma)[H] \cap N[H]$ with the hypothesized properties with respect to $G \cap (M \cap M^* \cap V_\gamma)[H]$ and the infimum of $p$ and $q$.  Then apply induction again to obtain a filter $G_2 \subseteq \mathbb P \cap R[H] \cap N[H]$ with the hypothesized properties with respect to $G_1$.  Note that $G_2 \cap V_\gamma[H] = G_1 \cap (R \cap V_\gamma)[H]$.  Now consider the forcing $\mathbb A \subseteq \mathbb P \cap M^*[H]$ below $q$ where we freeze the decorations on $R$; this is clearly a projection of $\mathbb P \cap M^*[H]$ below $q$.  For any dense open $D \subseteq \mathbb A$ in $N[H \cap M^*]$, there is $q' \leq q$ in $D$ such that $q' \restriction (M^* \cap V_\gamma) \in G_1$.  Then since $q' \restriction (R \cap V_\gamma) \in G_2$ and $G_2$ is generic for the quotient $(\mathbb P \cap R[H]) / (G_2 \cap V_\gamma[H])$, there is $q'' \leq q' \restriction R[H]$ such that $q'' \in D \restriction R[H]$ and $q'' \in G_2$.  Then $q'$ and $q''$ have an infimum $q'''$ in $\mathbb A$.  Note that since $q'' \restriction V_\gamma \in G_1$, $q''' \restriction (M^* \cap V_\gamma) \in G_1$.  So if $G_3$ is the filter of conditions $t \in \mathbb A$ such that $t \restriction (M^* \cap V_\gamma) \in G_1$ and $t \restriction R \in G_2$, then $G_3$ is $\mathbb A$-generic over $N[H \cap M^*]$.
The remaining quotient is just the decoration attached to $R$, and this is equivalent to the forcing $\col(\omega_1,|M^*|)^{V[H \cap M^*]}$.  We can build a generic for this over $N[H \cap M^*][G_3]$.
If $G^*$ is the generated filter, it is $(\mathbb P \cap M^*[H])$-generic over $N[H \cap M^*]$, and thus fully generic over $N[H]$ by Lemma \ref{magic}.
It is a member of $V[H \cap M^*]$.  The set $m_{G^*}$ is a condition because $m_{G^*} \restriction (M^* \cap V_\gamma) = m_{G_1}$, $m_{G^*} \restriction R = m_{G_2}$, and every model appearing in $G^*$ appears in either $G_1$ or $G_2$.

\textbf{Case 2.2.2:} $V_\gamma \notin \mathcal{T}$.

Let $\langle W_i : i < \omega \rangle \subseteq \mathcal{T} \cap N$ be cofinal in $V_\gamma \cap N$.  We can assume the sequence is a member of $V$.
Use the induction hypothesis, with respect to $G \cap (R \cap M)[H]$ and the infimum of $p$ and $q$, to obtain a filter $G' \subseteq \mathbb P \cap R[H] \cap N[H]$ with the hypothesized properties.
In particular, $G' \cap (R \cap M \cap W_i)[H] = G \cap (R \cap M \cap W_i)[H]$ for each $i$.

Let $\langle D_i : i < \omega \rangle$ enumerate the dense open subsets of $\mathbb P \cap M^*[H]$ that live in $N[H \cap M^*]$.
Let $p_0 \in D_0$ be stronger than $q$ and such that $p_0 \restriction R \in G'$.
Then use the induction hypothesis to find a generic filter $G_0 \subseteq \mathbb P \cap (M^* \cap W_0)[H] \cap N[H]$ extending $G \cap (M \cap M^* \cap W_0)[H]$, such that $p_0 \restriction (M^* \cap W_0) \in G_0$ and $m_{G_0}$ is a condition in $V[H \cap M^*]$.
Then let $p'_1 \in D_1$ be stronger than $p_0$ and such that $p'_1 \restriction (M^* \cap W_0) \in G_0$.  Since $G'$ is generic for the quotient
 $(\mathbb P \cap R[H]) / (G_0 \cap (R \cap W_0)[H])$, there is $p''_1 \leq p_1' \restriction R$ such that $p''_1 \in D_1 \restriction R[H]$ and $p''_1 \restriction R \in G'$.
 If $p_1$ is the infimum of $p_1'$ and $p_1''$, then $p_1 \restriction (M^* \cap W_0) \in G_0$ and $p_1 \restriction R \in G'$.
Continue in this way, producing a descending sequence of conditions $p_i \in D_i$ and an ascending sequence of generic filters $G_i \subseteq \mathbb P \cap (M^* \cap W_i)[H] \cap N[H]$ such that $p_i \restriction R \in G'$, $p_i \restriction (M^* \cap W_i) \in G_i$, and $m_{G_i}$ is a condition in $V[H \cap M^*]$.

If $G^*$ is the filter generated by the sequence of $p_i$'s, then $G^* \in V[H \cap M^*]$, and $G^*$ is generic for $\mathbb P \cap M^*[H]$ over $N[H]$.  As in \Coffeecup, $m_{G^*} \restriction R = m_{G'}$, and for each $i<\omega$, $m_{G^*} \restriction (M^* \cap W_i) = m_{G_i}$.  $m_{G^*}$ is a condition because each model that appears in $G^*$ either appears in $G'$ or in some $G_i$.

\textbf{Case 2.3:}  There is $q \leq p \restriction M^*$ in $M^*[H]$ that has no largest-rank model, and $q$ cannot be extended to a stronger condition in $M^*[H]$ with a model above all of the models of $q$.

\textbf{Case 2.3.1:} $V_\gamma \in \mathcal{T}$.

Use the induction hypothesis to obtain a filter $G' \subseteq \mathbb P \cap (M^* \cap V_\gamma)[H] \cap N[H]$ with the hypothesized properties with respect to the infimum of $p$ and $q$ and the filter $G \cap (M \cap M^* \cap V_\gamma)[H]$.
Let $\langle R_i : i < \omega \rangle$ be cofinal in the models of $q$, with $\gamma,\rank(M) <\rank(R_i)$ for each $i$.
Let $\langle D_i : i < \omega \rangle$ enumerate the dense open subsets of $\mathbb P \cap M^*[H]$ that live in $N[H \cap M^*]$.
Let $p_0 \in D_0$ be such that $p_0 \leq q$ and $p_0 \restriction (M^* \cap V_\gamma) \in G'$.
By induction, let $G_0 \subseteq \mathbb P \cap R_0[H] \cap N[H]$ be a generic filter such that $p_0 \restriction R_0 \in G_0$, $G_0 \cap (M \cap R_0 \cap V_\gamma)[H] = G' \cap (M \cap R_0 \cap V_\gamma)[H]$, and $m_{G_0}$ is a condition in $V[H \cap R_0]$.
Since $G'$ is generic for the quotient $(\mathbb P \cap (M^* \cap V_\gamma)[H]) /  (G_0 \cap V_\gamma[H])$, let $p_1 \leq p_0$ be in $D_1$ and such that $p_1 \restriction (M^* \cap V_\gamma) \in G'$.  Then use the hypothesis again to pick $G_1 \subseteq \mathbb P \cap R_1[H] \cap N[H]$ such that $p_1 \restriction R_1 \in G_1$ and the other desired properties hold.  Continue in this way, producing a descending sequence of conditions $p_i \in D_i$ and an ascending sequence of generic filters $G_i \subseteq \mathbb P \cap R_i[H] \cap N[H]$ such that $p_i \restriction (M^* \cap V_\gamma) \in G'$, $p_i \restriction R_i \in G_i$, and $m_{G_i}$ is a condition in $V[H \cap R_i]$.  Just as before, if $G^*$ is the generated filter, then $G^* \in V[H \cap M^*]$, $G^*$ is sufficiently generic, and $m_{G^*}$ is a condition because $m_{G^*} \restriction R_i = m_{G_i}$, $m_{G^*} \restriction (M^* \cap V_\gamma) = m_{G'}$, and each model appearing in $G^*$ appears either in $G'$ or in some $G_i$.

\textbf{Case 2.3.2:} $V_\gamma \notin \mathcal{T}$.

First suppose that $M \in V_\gamma$.
Let $\langle W_i : i < \omega \rangle \subseteq \mathcal{T} \cap N$ be cofinal in $V_\gamma \cap N$ with $M \in W_0$, and let $\langle R_i : i < \omega \rangle$ be cofinal in the models of $q$, with $\gamma\leq\rank(R_i)$ for each $i$.
Let $\langle D_i : i < \omega \rangle$ enumerate the dense open subsets of $\mathbb P \cap M^*[H]$ that live in $N[H \cap M^*]$.
First use the induction hypothesis with respect to $G \cap (M \cap M^* \cap W_0)[H]$ and the infimum of $p$ and $q$, obtaining a filter $G^*_0 \subseteq \mathbb P \cap (M^* \cap W_0)[H]$ with $p_0^* \restriction (M^* \cap W_0) \in G^*_0$, where $p_0^* \in D_0$ and $p_0^* \leq q$.  Then apply the hypothesis with respect to $G^*_0$ and the model $R_0$ to obtain a generic filter $G_0 \subseteq \mathbb P \cap R_0[H]$ such that $G_0 \cap W_0[H] \subseteq G_0^*$.  Using the fact that $G_0$ is generic for the quotient $(\mathbb P \cap R_0[H]) / (G_0 \cap W_0[H])$, find $p'_0 \leq p_0^* \restriction R_0$ in $D_0 \restriction R_0[H]$ such that $p'_0 \in G_0$.  If $p_0$ is the infimum of $p_0^*$ and $p_0'$, then $p_0 \restriction (M^* \cap W_0) \in G_0^*$ and $p_0 \restriction R_0 \in G_0$.
Continue in this way, producing a descending sequence of conditions $p_i \in D_i$ and sequences of filters $G_i^*$ and $G_i$ such that $G_i^*$ is generic for $\mathbb P \cap (M^* \cap W_i)[H]$, $G_i$ is generic for $\mathbb P \cap R_i[H]$, $G_i \cap W_i[H] \subseteq G_i^*$, and $p_i \restriction W_i \in G_i^*$ and $p_i \restriction R_i \in G_i$.  If $G^*$ is the generated filter, then $G^* \in V[H \cap M^*]$, and $G^* \cap (M^* \cap M)[H] = G^*_0 \cap (M\cap M^*)[H] = G \cap (M \cap M^*)[H]$.  As before, $m_{G^*}$ is a condition because $m_{G^*} \restriction R_i = m_{G_i}$, $m_{G^*} \restriction (M^* \cap W_i) = m_{G_i^*}$, and every model that appears in $G^*$ appears either in some $G_i$ or some $G_i^*$.

If $\rank(M) \geq \gamma$, the situation is just slightly different.  Let $\langle W_i : i < \omega \rangle \subseteq \mathcal{T} \cap N$ be cofinal in $V_\gamma \cap N$, and let $\langle R_i : i < \omega \rangle$ be cofinal in the models of $q$, with $R_0 = M$.  Then repeat the same construction.  Here we have $G^* \cap (M \cap M^*)[H] = G^* \cap R_0[H] = G$.
\end{proof}

\begin{corollary}
In $V[H]$, for any $p \in \mathbb P \cap N[H]$, there is $m \leq p$ that generates a $\mathbb P$-generic filter over $N[H]$.  Thus, since $N$ was any sufficiently elementary countable model, $\mathbb P$ adds no $\omega$-sequences of ordinals.
\end{corollary}

\begin{proof}
Since $\mathcal{S}[H]$ is stationary, for any dense set $D \subseteq \mathbb P$, there are stationary-many $M[H] \in \mathcal{S}[H]$ such that $D \cap M[H]$ is dense in $\mathbb P \cap M[H]$.  This is known to $N[H]$.  Let $\langle D_i : i < \omega \rangle$ enumerate the dense subsets of $\mathbb P$ that live in $N[H]$.  Trivially, $\emptyset$ is a filter that is $(N[H], \mathbb P \cap M[H])$-generic, where $M$ is an $\in$-minimal member of $\mathcal{S}$.  By the above observation, for any $p \in \mathbb P \cap N[H]$, there is there is $M_0 \in \mathcal{S} \cap N$ and a $p_0 \leq p$ such that $p_0 \in D_0 \cap M_0[H]$.  Applying Lemma \ref{genericsexist}, there is a filter $G_0$ that is $(N[H], \mathbb P \cap M_0[H])$-generic with $p_0 \in G_0$, and such that $m_{G_0}$ is a condition in $M_0[H]$.  Continuing this process, we obtain a descending sequence of conditions $p_i$ and ascending sequences of models $M_i \in \mathcal{S} \cap N$ and filters $G_i$ such that $p_i \in D_i \cap M_i[H] \cap G_i$.  As in the argument for Lemma \ref{genericsexist}, there is a lower bound $p_\omega$ that generates an $(N[H],\mathbb P)$-generic filter.
\end{proof}

Now let us argue for the tree property at $\kappa$.  Towards a contradiction, let $\dot T$ be an $\add(\omega,\kappa) * \dot{\mathbb P}$-name for a $\kappa$-tree that is forced to be Aronszajn.
Let $\langle d,\dot e\rangle \in \add(\omega,\kappa) * \dot{\mathbb P}$ be arbitrary.
Let $\theta^*>\theta$ be regular, let $\lhd^*$ be a well-order on $H_{\theta^*}$, and let $M^* \prec \frak B = \langle H_{\theta^*},\in,\lhd^*,\frak A,\dot T,\langle d,\dot e \rangle \rangle$ and such that $M = M^* \cap V_\theta \in \mathcal{S}$ and $\ot(M \cap \theta)$ is inaccessible.  Let $N \prec \frak B$ be countable with $M^* \in N$.

Let $\pi : M^* \to R$ be the transitive collapse.  For $x\in M^*$, let $\bar x$ denote $\pi(x)$.
We claim that $\langle \omega_1,\bar\kappa,\bar\theta,\langle V_\alpha : \alpha < \bar\theta \rangle,\bar\mathcal{S},\bar\mathcal{T} \rangle$ is nice and hence $\bar{\mathbb P}$ is forced to have the $\bar\theta$-c.c.  If $\bar\alpha < \gamma$ is such that $\cf(\bar\alpha) \geq \bar\kappa$ and $V_{\bar\alpha} \prec \bar{\frak A}$, then $\cf(\alpha) \geq \kappa$ and $V_\alpha \prec \frak A$.  Thus $\bar\mathcal{T}$ contains all elementary ranks of high cofinality, so the claim is shown.

By Lemma \ref{genericsexist}, let $\dot m$ be an $\add(\omega,\bar\kappa)$-name for a condition below $\dot e$ in $\dot{\mathbb P} \cap M[\dot H]$ that generates a filter on $\dot{\mathbb P} \cap M[\dot H]$ that is generic over $N[\dot H]$.  Let $\dot m'$ be a name for an extension of $\dot m$ that includes the model $M$.  Force below $\langle d,\dot m' \rangle$ to obtain a generic filter $H * G$.  Let $h * g = (H * G) \cap M$, which is $\add(\omega,\bar\kappa) * (\dot{\mathbb P} \cap M[\dot H])$-generic over $V$ by the fact that $\langle 1,\dot m' \rangle$ is a strong master condition for $M^*$.

Let $T = \dot T^{H * G}$ and let $t = T \cap M[H * G]$, which can be computed in $V[h * g]$.  We claim that $t$ is an Aronszajn $\bar\kappa$-tree in $V[h*g]$.  Its levels have size ${<}\bar\kappa$ by the elementarity of $M^*[H * G]$ in $H_{\theta^*}[H * G]$.  If there were a cofinal branch $b \in V[h*g]$, then since $\add(\omega,\bar\kappa) * \bar{\dot{\mathbb P}}$ is $\bar\theta$-c.c., there would be a name $\dot b$ for such a branch in $V_{\bar\theta}$.  But then $\pi^{-1}(\dot b) \in M$ would be a name for a cofinal branch of $\dot T$, contradicting that $\dot T$ is forced to be Aronszajn.  The same argument also shows that $\bar\kappa = \omega_2$ in $V[g*h]$.
However, since $t$ is an initial segment of $T$, it does acquire a cofinal branch in $V[H*G]$.
Let $\dot b \in N[H * g]$ be a name for such a branch.

Let $\mathbb Q$ be the quotient $(\add(\omega,\kappa) * \dot{\mathbb P} )/ (h*g)$.
By Lemma \ref{posetmeetmodel}, $\mathbb P \cap N[H]$ can be computed in $V[H \cap N]$.
Let $g_\emptyset = N[H] \cap g$.  Then $g_\emptyset$ is exactly the filter generated by $\dot m^H$.
Let $\sup(N \cap \kappa) < \delta < \kappa$.
Using Lemma \ref{genericsexist}, working in $V[H \cap V_\delta]$, we can build the following copy of the tree $\omega^{<\omega}$.
First let $\langle q_i : i < \omega \rangle$ enumerate the elements of $\mathbb Q \cap N[H]$.
Let $\{ p_{\langle i \rangle} : i < \omega \}$ be a maximal antichain of elements in which $M$ appears such that one of them is below $q_0$.
Let $M_{\langle i \rangle} \in N \cap \mathcal{S}$ be such that $p_{\langle i \rangle} \in M_{\langle i \rangle}[H]$.
Then for each $i$, take a filter $g_{\langle i \rangle}$ extending $g_\emptyset$ that is $(N[H], \mathbb P \cap M_{\langle i \rangle}[H])$-generic and such that $m_{g_{\langle i \rangle}}$ is a condition in $M_{\langle i \rangle}[H]$.
Continue in this way, producing a tree of conditions, models, and filters indexed by $\sigma \in \omega^{<\omega}$, such that for each $i$, $q_i \geq p_{\sigma}$ for some $\sigma$ of length $i+1$, and for each $\sigma$, $p_\sigma \in g_\sigma \in M_\sigma[H]$, $m_{g_\sigma}$ is a condition in $M_\sigma[H]$, and $M_{\sigma \restriction i}$ occurs in $p_\sigma$ for $i < \len(\sigma)$.

Now let $r \in \omega^\omega$ be a Cohen-generic real appearing in $V[H] \setminus V[H \cap V_\delta]$.
If use use $r$ to follow the path through our through our copy of $\omega^{<\omega}$, then this generates a filter $g_r$ over $\mathbb P \cap N[H]$.
For any dense open $D \subseteq \mathbb Q \cap N[H]$ living in $V[H \cap V_\delta]$, $g_r$ meets $D$, since any $\sigma \in \omega^{<\omega}$ can be extended to some $\sigma'$ such that $p_{\sigma'} \in D$.
Moreover, $g_r$ determines a condition $m_{g_r}$ in $\mathbb P$.  This is because if $R$ is a model appearing in $m_{g_r}$, then there is $n<\omega$ such that $R \in M_{r \restriction n}$ and thus $R$ already appears in $m_{g_{r \restriction n}}$.  Since $m_{g_{r \restriction n}} = m_{g_r} \restriction M_{r \restriction n}$, we have $m_{g_r} \restriction R =  m_{g_{r \restriction n}} \restriction R$, which is in $R[H]$ since $m_{g_{r \restriction n}}$ is a condition.

Thus, ignoring the original choice of $G$, if we force below $m_{g_r} \in \mathbb P/g$ we have that $(H * G) \cap N$ is $(\mathbb Q \cap N[h*g])$-generic over $V[g*h]$.  By the basic theory of strongly proper forcing (see \cite {mv} or \cite{neeman}), there is a strong master condition $q \in \mathbb Q$ for $N[h*g]$.  This means that for every $x \leq q$, there is a condition $x_N \in N[h*g]$ such that every $y \leq x_N$ in $N[h*g]$ is compatible with $x$.  Now since $t$ is a tree in $V[h*g]$ and $N$ is countable, there is $x \leq q$ that completely decides $N[h*g] \cap \dot b$ by deciding the node of $\dot b$ at level $\sup(N \cap M \cap \kappa)$.  However, because $\dot b$ is forced to be a new set, the elementarity of $N[h*g]$ gives that there are $y_0,y_1 \leq x_N$ that decide contradictory information about $\dot b$.  But this is a contradiction since they are both compatible with $x$.  This concludes the proof of Theorem \ref{tpthm}.
\end{proof}

\subsection{Limitative results}
\label{limitations}

Combining Theorems \ref{forcewcc} and \ref{tpthm}, we get:

\begin{corollary}
\label{combi}
If $\kappa<\theta$ are inaccessible, $\kappa$ is $\theta$-supercompact, and $V_\theta \models$ ``$\kappa$ is ambitious'', then there is a forcing extension satisfying, $2^\omega= \omega_2$, the tree property at $\omega_2$, and $\wcc(\omega_2,\cof(\omega_1))$.
\end{corollary}

Let us recall the main result of \cite{coxeskew}:

\begin{theorem}
\label{coxeskewthm}
Suppose $\kappa$ is a regular cardinal and $2^{<\kappa} \leq \kappa^+$.  If there is a weakly presaturated ideal $\ensuremath{\mathcal{I}}$ on $\kappa^+$ such that $\kappa^+ \cap \cof(\kappa) \in \ensuremath{\mathcal{I}}^d$, then there is a special Aronszajn $\kappa^+$-tree.
\end{theorem}

This implies that in an extension given by Corollary \ref{combi}, there is no weakly presaturated ideal on $\omega_2$ concentrating on $\cof(\omega_1)$.  Thus the hypothesis of the existence of such an ideal cannot be weakened to  $\wcc(\omega_2,\cof(\omega_1))$ in Theorem \ref{coxeskewthm}.

The author tried to find some large cardinal hypothesis that would entail the existence of a stationary set of Magidor models with which one could mimic the structure of the proof of Theorem \ref{wps-ee} and obtain a weakly presaturated ideal on $\omega_2$ along with the tree property.  Eventually, he realized that Theorem \ref{tpthm} is indirectly revealing a ZFC-constraint that blocks such approaches.

\begin{theorem}
\label{shortmag}
Suppose $\kappa$ is $\theta$-supercompact, where $\theta > \kappa$ is inaccessible.  Then for every stationary set $\mathcal{S} \subseteq \mathcal{P}_\kappa(V_\theta)$ of $\kappa$-Magidor models $M$ that are ${<}(M\cap\kappa)$-closed, there is:
\begin{itemize}
\item a structure $\frak B$ on $V_\theta$ in a countable language,
\item an ordinal $\delta<\theta$,
\item a set $\mathcal{D} \subseteq \mathcal{P}_\kappa(V_\delta)$, and
\item a function $b : \mathcal{D} \to \kappa$
\end{itemize}
such that:
\begin{itemize}
\item $\{ M \in \mathcal{S} : M \cap V_\delta \in \mathcal{D}\}$ is stationary, and
\item if $M \in \mathcal{S}$, $M \prec \frak B$, and $M \cap V_\delta \in \mathcal{D}$, then $\ot(M \cap \theta) < b(M \cap V_\delta)$.
\end{itemize}
\end{theorem}

\begin{proof}
Let $\kappa,\theta,\mathcal{S}$ be as hypothesized.  Let $\frak A$ be a structure on $V_\theta$ in a countable language that includes a well-order.  Since $\add(\omega,\kappa)$ preserves stationarity, Lemma \ref{statpres} implies that if $H*G \subseteq \add(\omega,\kappa) * \dot{\mathbb M}^{\frak A}(\omega_1,\kappa,\theta)$ is generic, then the set of $M \in \mathcal{S}$ that appear in $G$, which we will denote here by $\mathcal{S}_G$, is stationary in $V[H*G]$.  In this universe, consider the ideal $\ensuremath{\mathcal{I}}$ on $\kappa = \omega_2$ that is the canonical projection of $\ns \restriction \mathcal{S}_G$, i.e.\ $X \in \ensuremath{\mathcal{I}}$ if and only if $\{ M \in \mathcal{S}_G : M \cap \kappa \in X \}$ is nonstationary.

By Theorem \ref{tpthm}, $V[H*G]$ satisfies the tree property at $\omega_2$ and $2^\omega = \omega_2$, and thus Theorem \ref{coxeskewthm} implies that $\ensuremath{\mathcal{I}}$ is not weakly presaturated.  Using Lemma \ref{wps-equivs}, this means that there is an $\ensuremath{\mathcal{I}}$-positive $B \subseteq \kappa$ and a function $f : \kappa \to \kappa$ such that $\{ M \in \mathcal{S}_G : M \cap \kappa \in B \wedge f(M \cap \kappa) < \ot(M \cap \theta) \}$ is nonstationary.  Let $F : \theta^{<\omega} \to \theta$ be such that any $M \in \mathcal{S}_G$ closed under $F$ is not in the above set.

In $V$, let $\dot B$, $\dot f$, and $\dot F$ be names for these objects.  By the $\theta$-c.c., there is $\delta<\theta$ with $V_\delta \in \mathcal{T}$ such that we can take $\dot B$ and $\dot f$ to be $\mathbb P_\delta$-names, where $\mathbb P_\delta$ is the intersection of the whole forcing with $V_\delta$.  Let $\theta^*>\theta$ be regular, and let $\frak B^* = \langle H_{\theta^*},\in,\kappa,\delta,\theta,\frak A,\mathcal{S},\dot B,\dot f,\dot F,\lhd \rangle$, where $\lhd$ is a well-order.  Let $\frak B$ be the result of restricting the Skolem functions for $\frak B^*$ to $V_\theta$.

Since $\dot B$ is forced to be $\ensuremath{\mathcal{I}}$-positive, it is forced that there are stationary-many $M \in \mathcal{S}_{\dot G}$ such that $M \cap \kappa \in \dot B$.  If $\langle q,\dot p \rangle$ forces ``$\alpha = M \cap \kappa \in \dot B$ and $M \in \mathcal{S}_{\dot G}$'', then we may assume $q$ forces that $\dot p$ includes the models $M$, $V_\delta$, and $M \cap V_\delta$.  Let $\mathcal{D}$ be the set of all $N \in \mathcal{P}_\kappa(V_\delta)$ such that
some $\langle q,\dot p \rangle \in \mathbb P_\delta$ has the property that $q$ forces $\dot p$ to include $N$ and $\langle q,\dot p \rangle$ forces $N \cap \kappa \in \dot B$.  Note that $\{ M \in \mathcal{S} : M \cap V_\delta \in \mathcal{D} \}$ must be stationary since it is forced to have a stationary subset in $V[H*G]$.

Suppose towards a contradiction that for some $N \in \mathcal{D}$, there are $M \in \mathcal{S}$ elementary in $\frak B$ with $\ot(M \cap \theta)$ arbitrarily large below $\kappa$ and such that $M \cap V_\delta = N$.  Let $\langle q,\dot p \rangle$ witness that $N \in \mathcal{D}$.  Let $\langle q',\dot p' \rangle \in \mathbb P_\delta$ strengthen $\langle q,\dot p \rangle$ and decide the value of $\dot f(N \cap \kappa)$, say as $\xi$.  Let $M \prec \frak B$ be in $\mathcal{S}$ and such that $M \cap V_\delta = N$ and $\ot(M \cap \theta)>\xi$.
If $\dot s'$ is forced to be the set of models appearing in $\dot p'$, then if $\dot s''$ is a name for $\dot s' \cup \{ V_\delta,M \}$, then just as in the proof of Theorem \ref{forcewcc}, $\dot s''$ is forced to be closed under intersections and have the property that $\dot s'' \cap M \in M[H]$.  Let $\dot p''$ be a name for the weakest condition below $\dot p'$ whose models are $\dot s''$, yielding $\langle q',\dot p'' \rangle \leq \langle q',\dot p' \rangle$.
Since $\sk^{\frak B^*}(M) \cap V_\theta = M$ and $\langle q',\dot p'' \rangle$ is a strong master condition for $M$,
$\langle q',\dot p'' \rangle$ forces that $M$ is closed under $\dot F$, $M \cap \kappa \in \dot B$, and $\ot(M \cap \theta) > \xi = \dot f(M \cap \kappa)$.  This contradicts the properties that $\dot B$, $\dot f$, and $\dot F$ are forced to satisfy.

Thus for all $N \in \mathcal{D}$, there is a bound $b(N)<\kappa$ on $\ot(M \cap \theta)$ for $M \in \mathcal{S}$ such that $M \prec \frak B$ and $M \cap V_\delta = N$, as claimed.
\end{proof}

\begin{question*}
Can we find a forcing-free proof of Theorem \ref{shortmag}?
\end{question*}

\begin{question*}
Can we strengthen Theorem \ref{shortmag} by replacing (a) $\theta$-supercompactness with ${<}\theta$-supercompactness, or (b) $\delta$ with $\kappa$?
\end{question*}

\begin{corollary}
Suppose $\kappa<\theta$ are inaccessible, $\kappa$ is $\theta$-supercompact, $\mu<\kappa$ is regular, and $\frak A = \langle V_\alpha,\in,\mu,\kappa \rangle$.
If $\mathcal{S}_0$ is a stationary set of $\kappa$-Magidor $M \prec \frak A$ that are ${<}(M \cap \kappa)$-closed, $\mathcal{T}$ is the set of all $V_\alpha\prec \frak A$ with $\cf(\alpha) \geq \kappa$, and $\mathcal{S} = \{ M \cap W : M \in \mathcal{S}_0 \wedge W \in \mathcal{T} \}$,
then $\mathbb P^{\mathrm{dec}*}_{\mu,\mathcal{S},\mathcal{T}}$ forces that $\ns_\kappa \restriction \cof(\mu)$ is not weakly presaturated.
\end{corollary}

\begin{proof}
Let $\frak B,\delta,\mathcal{D},b$ be given by Theorem \ref{shortmag} with respect to $\mathcal{S}$.  Let $W = V_\delta$ and let $G \subseteq \mathbb P^{\mathrm{dec}*}_{\mu,\mathcal{S},\mathcal{T}}$ be generic.  Then $\mathcal{D}$ is a stationary subset of $\mathcal{P}_\kappa(W)$ in $V[G]$.   In $V[G]$, let $f : \kappa \to W$ be a bijection.  Let $D = \{ \alpha < \kappa : f[\alpha] \in \mathcal{D}$ and $f[\alpha] \cap \kappa = \alpha \}$.  Then $D$ is a stationary subset of $\kappa \cap \cof(\mu)$.

By Lemma \ref{*proj}, $\ns_\kappa \restriction \cof(\mu)$ is the canonical projection of $\ns \restriction \mathcal{S}_G$, where $\mathcal{S}_G$ is the set of $M \in \mathcal{S}$ appearing in $G$.  By Lemma \ref{wps-equivs}, it suffices to show that there is a function $g : \kappa \to \kappa$ such that $\{ M \in \mathcal{S}_G : M \cap \kappa \in D  \wedge g(M \cap \kappa) < \ot(M \cap \theta) \}$ is nonstationary.  For $\alpha \in D$, let $g(\alpha) = b(f[\alpha])$, and otherwise let $g(\alpha) = 0$.  If $M \in \mathcal{S}_G$ is closed under $f$ and $f^{-1}$, then $M \cap W = f[M \cap \kappa]$.  If additionally $M \prec \frak B$ and $M \cap \kappa \in D$, then $\ot(M \cap \theta) < b(M \cap W) = g(M \cap \kappa)$.
\end{proof}

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redpajama_set_name: RedPajamaArXiv