text:
\section{Introduction}

For an integer $k \geq 2$ and a $k$-uniform hypergraph $H$, the \emph{Ramsey number} $R(H)$ is the minimum $N \in \mathbb{N}$ such that every 2-coloring of the hyperedges of the complete $k$-uniform hypergraph $K^{(k)}_N$ on~$N$ vertices contains a monochromatic subhypergraph isomorphic to $H$.
Estimating Ramsey numbers is a notoriously difficult problem.
Despite many efforts in the last 70 years, no tight bounds are known even for the complete graph $K_n$ on $n$ vertices.
Apart from some smaller term improvements, essentially the best known bounds are $2^{n/2} \leq R(K_n) \leq 2^{2n}$ by Erd\H{o}s~\cite{erdosSzekeres35} and by Erd\H{o}s and Szekeres~\cite{erdos47}.
The Ramsey numbers $R(K^{(k)}_n)$ are even less understood for $k \geq 3$.
For example, it is only known that
\begin{equation}
\label{eq-hypergraphRamsey}
2^{\Omega(n^2)} \leq R(K^{(3)}_n) \leq 2^{2^{O(n)}},
\end{equation}
as shown by Erd\H{o}s, Hajnal, and Rado~\cite{erdosRado65}.
A famous conjecture of Erd\H{o}s, for whose proof Erd\H{o}s offered \$500 reward, states that there is a constant $c>0$ such that $R(K^{(3)}_n) \geq 2^{2^{cn}}$.

Recently, a variant of Ramsey numbers for hypergraphs with a fixed order on their vertex sets has been introduced~\cite{bckk13,clfs17}.
For an integer $k \geq 2$, an \emph{ordered $k$-uniform hypergraph} $\mathcal{H}$ is a pair $(H,<)$ consisting of a $k$-uniform hypergraph $H$ and a linear ordering $<$ of its vertex set.
An ordered $k$-uniform hypergraph $\mathcal{H}_1=(H_1,<_1)$ is an \emph{ordered subhypergraph} of another ordered $k$-uniform hypergraph $\mathcal{H}_2=(H_2,<_2)$, written $\mathcal{H}_1 \subseteq \mathcal{H}_2$, if $H_1$ is a subhypergraph of $H_2$ and $<_1$ is a suborder of $<_2$.
Two ordered hypergraphs $\mathcal{H}_1$ and $\mathcal{H}_2$ are \emph{isomorphic} if there is an isomorphism between their underlying hypergraphs that preserves the vertex orderings of $\mathcal{H}_1$ and $\mathcal{H}_2$.
Note that, up to isomorphism, there is a unique ordered complete $k$-uniform hypergraph $\mathcal{K}^{(k)}_n$ on $n$ vertices.

The \emph{ordered Ramsey number} $\overline{R}(\mathcal{H},\mathcal{G})$ of two ordered $k$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{G}$ is the smallest $N \in \mathbb{N}$ such that every coloring of the hyperedges of $\mathcal{K}^{(k)}_N$ by colors red and blue contains a blue ordered subhypergraph isomorphic to $\mathcal{H}$ or a red ordered subhypergraph isomorphic to $\mathcal{G}$.
In the \emph{diagonal case} $\mathcal{H} = \mathcal{G}$, we just write $\overline{R}(\mathcal{G})$ instead of $\overline{R}(\mathcal{G},\mathcal{G})$.

The ordered Ramsey numbers are known to be finite and it is easy to see that they grow at least as fast as the standard Ramsey numbers.
Studying ordered Ramsey numbers has attracted a lot of attention lately (see the survey by Conlon, Fox, and Sudakov~\cite{cfsSurvey}), as there are various motivations coming from the field of discrete geometry.
It is known that ordered Ramsey numbers can behave quite differently than the standard Ramsey numbers, especially for sparse ordered graphs~\cite{bckk13,bjv16,clfs17}.
However, so far, the ordered Ramsey numbers have been studied mostly for ordered graphs only and very little is known about ordered Ramsey numbers of ordered $k$-uniform hypergraphs with $k \geq 3$.

In this paper, we focus on $3$-uniform hypergraphs and we prove some new bounds on the ordered Ramsey numbers of ordered tripartite $3$-uniform hypergraphs.
We also pose several new open problems in Section~\ref{sec-openProblems}.

\subsection{Preliminaries}

For an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ and two subsets $U$ and $V$ of vertices of~$\mathcal{H}$, we say that $U$ and $V$ are \emph{consecutive} if all vertices from $U$ precede all vertices of $V$ in~$<$.
An \emph{interval} in $\mathcal{H}$ is a subset $I$ of vertices of $\mathcal{H}$ such that for all vertices $u,v,w$ of $\mathcal{H}$ with $u<v<w$ and $u,w \in I$ we have $v \in I$.

For integers $k \geq 2$ and $\chi \geq k$, we use $K^{(k)}_\chi(n)$ to denote the \emph{complete $k$-uniform $\chi$-partite hypergraph}, that is, the vertex set of $K^{(k)}_\chi(n)$ is partitioned into $\chi$ sets of size $n$ and every $k$-tuple with at most one vertex in each of these parts forms a hyperedge.
The ordering of~$K^{(k)}_\chi(n)$, in which the color classes form consecutive intervals, is denoted by $\mathcal{K}^{(k)}_\chi(n)$.
We sometimes use $\mathcal{K}_{n,n}$ to denote $\mathcal{K}^{(2)}_2(n)$.

The \emph{degree} of a vertex $v$ in a hypergraph $H$ is the number of hyperedges of $H$ that contain~$v$.
For $d \in \mathbb{N}$, a $k$-uniform hypergraph $H$ is \emph{$d$-degenerate} if there is an ordering $v_1 \prec \cdots \prec v_t$ of vertices of $H$ such that each $v_i$ is contained in at most $d$ hyperedges of $H$ that contain a vertex from $v_1,\dots,v_{i-1}$.

For a positive integer $n$, we use $[n]$ to denote the set $\{1,\dots,n\}$.
We omit floor and ceiling sign whenever they are not crucial and we use $\log$ and $\ln$ to denote base $2$ logarithm and the natural logarithm, respectively.

\subsection{Previous results}
\label{subsec-previousResults}

The ordered Ramsey numbers of $k$-uniform ordered hypergraphs with $k \geq 3$ remain quite unexplored.
Only the ordered Ramsey numbers of so-called monotone hyperpaths are well understood due to their close connections to the famous Erd\H{o}s--Szekeres Theorem~\cite{erdosSzekeres35}; see~\cite{balko19,fpss12,moshShap14}.
A \emph{monotone hyperpath} $\mathcal{P}^{(k)}_n$ on $n$ vertices is an ordered $k$-uniform hypergraph where the hyperedges are formed by $k$-tuples of consecutive vertices.
Note that the maximum degree of a $k$-uniform monotone hyperpath is at most $k$.
Moshkovitz and Shapira~\cite{moshShap14} showed that $\overline{R}(\mathcal{P}^{(k)}_n) = {\rm tow}_{k-1}((2-o(1))n)$ for $k \geq 3$, where ${\rm tow}_h$ is the \emph{tower function} of height $h$ defined as ${\rm tow}_1(x)=x$ and ${\rm tow}_h(x) = 2^{{\rm tow}_{h-1}(x)}$ for $h \geq 2$.

Thus even for $3$-uniform hypergraphs $\mathcal{H}$ with bounded maximum degree the numbers $\overline{R}(\mathcal{H})$ can grow very fast.
We get an exponential lower bound on $\overline{R}(\mathcal{H})$ even for $3$-uniform ordered hypergraphs $\mathcal{H}$ with maximum degree $3$.
A similar result is known for ordered graphs, as for arbitrarily large values of $n$ there are ordered graphs $\mathcal{M}_n$ with $n$ vertices and maximum degree $1$ such that $\overline{R}(\mathcal{M}_n) \geq n^{\Omega(\log{n}/\log{\log{n}})}$~\cite{bckk13,clfs17}.
This superpolynomial growth rate is in sharp contrast with  the situation for unordered hypergraphs, where the Ramsey number $R(H)$ of every $k$-uniform hypergraph $H$ with bounded $k$ and with bounded maximum degree is at most linear in the number of vertices of~$H$~\cite{crst83,cfs09,cnko08,cnko09,ishingami07,nsrs08}.

Therefore, in order to obtain smaller upper bounds on the ordered Ramsey numbers, it is necessary to bound other parameter besides the maximum degree.
A natural choice is so-called interval chromatic number, which can be understood as an analogue of the chromatic number for ordered graphs due to a variant of the Erd\H{o}s--Stone--Simonovits theorem for ordered graphs proved by Pach and Tardos~\cite{pachTardos06}.
The \emph{interval chromatic number}
$\chi_<(\mathcal{H})$ of an ordered $k$-uniform hypergraph $\mathcal{H}$ is the minimum number of intervals the vertex set of $\mathcal{H}$ can be partitioned into so that each hyperedge of $\mathcal{H}$ has at most one vertex in each of the intervals.

For ordered graphs, bounding both parameters indeed helps, as the ordered Ramsey number $\overline{R}(\mathcal{G})$ of every ordered graph $\mathcal{G}$ with bounded maximum degree $d$ and bounded interval chromatic number $\chi$ is at most polynomial in the number of vertices~\cite{bckk13,clfs17}.
Since $\mathcal{G} \subseteq \mathcal{K}^{(2)}_\chi (n)$, this result follows from the following stronger estimate proved by Conlon, Fox, Lee, and Sudakov~\cite{clfs17}: for all $d, \chi \in \mathbb{N}$, every $d$-degenerate ordered graph $\mathcal{G}$ on $n$ vertices with interval chromatic number $\chi$ satisfies
\begin{equation}
\label{eq-degreeIntChrNumber}
\overline{R}(\mathcal{G},\mathcal{K}^{(2)}_\chi(n)) \leq n^{32 d \log{\chi}}.
\end{equation}

A natural question is whether we can also get some good upper bounds on ordered Ramsey numbers of similarly restricted classes of ordered hypergraphs.
If the interval chromatic number is bounded, then we can use a result of Conlon, Fox, and Sudakov~\cite{cfs11}, who showed that, for all positive integers $\chi \geq 3$ and $n$, \[R(K^{(3)}_\chi(n)) \leq 2^{2^{2R}n^2},\]
where $R = R(K_{\chi-1})$.
Since every ordering of $K^{(3)}_\chi(\chi n)$ contains an ordered subhypergraph isomorphic to $\mathcal{K}^{(3)}_\chi(n)$ and every ordered $3$-uniform hypergraph on $n$ vertices with interval chromatic number $\chi$ is an ordered subhypergraph of $\mathcal{K}^{(3)}_\chi(n)$, we obtain the following bound.

\begin{corollary}
\label{cor-3UnifBoundedIntChr}
For all positive integers $\chi \geq 3$ and $n$, every ordered $3$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with interval chromatic number $\chi$ satisfies
\[\overline{R}(\mathcal{H}) \leq 2^{2^{2R}\chi^2n^2},\]
where $R = R(K_{\chi-1})$.
In particular, if the interval chromatic number $\chi$ of $\mathcal{H}$ is fixed, we have
\[
\overline{R}(\mathcal{H}) \leq 2^{O(n^2)}.
\]
\end{corollary}

Note that the last bound is asymptotically tight for dense ordered hypergraphs, as a standard probabilistic argument shows that $\overline{R}(\mathcal{H}) \geq 2^{\Omega(n^2)}$ for every ordered $3$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with $\Omega(n^3)$ hyperedges.
In particular, we get $\overline{R}(\mathcal{K}^{(3)}_3(n)) \geq 2^{\Omega(n^2)}$.

\section{Our results}

Since the bounds on the ordered Ramsey numbers from Corollary~\ref{cor-3UnifBoundedIntChr} are asymptotically tight for dense ordered hypergraphs with bounded interval chromatic number, we consider the sparse case with bounded maximum degree and interval chromatic number.
The situation for ordered hypergraphs seems to be more difficult than for ordered graphs, so we focus on the first nontrivial case, which is for ordered $3$-uniform hypergraphs with interval chromatic number $3$.

Assuming the maximum degree of an ordered hypergraph $\mathcal{H}$ with $\chi_<(\mathcal{H})=3$ is sufficiently small, we obtain a better upper bound on $\overline{R}(\mathcal{H})$ than the estimate $2^{O(n^2)}$ we would get from Corollary~\ref{cor-3UnifBoundedIntChr}.
We can prove an estimate with  a subquadratic exponent even in the more general setting $\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n))$, where, additionally, the interval chromatic number of $\mathcal{H}$ is arbitrary.

\begin{theorem}
\label{thm-3UnifMaxDegIntChr}
Let $\mathcal{H}$ be an ordered 3-uniform hypergraph on $t$ vertices with maximum degree $d$ and let $s$ be a positive integer.
Then there are constants $C=C(d)$ and $c>0$ such that
\[\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(s)) \leq t \cdot 2^{C(s^{2-1/(1+cd^2)})}.\]
In particular, for $s=t=n$ and bounded $d$, we get the estimate
\begin{equation}
\label{eq-diagonal}
\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n)) \leq 2^{O(n^{2-1/(1+cd^2)})}.
\end{equation}
\end{theorem}

The main idea of the proof of Theorem~\ref{thm-3UnifMaxDegIntChr} is based on an embedding lemma from~\cite{cfs12}, where the authors study Erd\H{o}s--Hajnal-type theorems for $3$-uniform tripartite hypergraphs.
We prove a variant of this lemma, which works for ordered hypergraphs, does not consider induced copies, and uses the assumption that the maximum degree of $\mathcal{H}$ is bounded instead of assuming that the number of vertices of $\mathcal{H}$ is fixed.

Since every ordered $3$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with $\chi_<(\mathcal{H})=3$ is an ordered subhypergraph of~$\mathcal{K}^{(3)}_3(n)$, we immediately obtain the following corollary.

\begin{corollary}
\label{cor-3UnifMaxDegIntChr}
Let $\mathcal{H}$ be an ordered 3-uniform hypergraph on $n$ vertices with maximum degree $d$ and with interval chromatic number $3$.
Then there exists an $\varepsilon = \varepsilon(d)>0$ such that
\[
\overline{R}(\mathcal{H}) \leq 2^{O(n^{2-\varepsilon})}.
\]
\end{corollary}

It might seem wasteful to use Theorem~\ref{thm-3UnifMaxDegIntChr} in order to obtain Corollary~\ref{cor-3UnifMaxDegIntChr}, as the ordered hypergraph $\mathcal{H}$ is much sparser than $\mathcal{K}^{(3)}_3(n)$.
However, as noted in~\cite{clfs17}, this intuition is wrong already for ordered graphs, as there are ordered matchings $\mathcal{M}$ on $n$ vertices with $\chi_<(\mathcal{M})=2$ and ordered graphs $\mathcal{G}$ on $N=2^{n^c}$ vertices for some constant $c>0$ such that $\mathcal{G}$ has edge density at least $1-n^{-c}$ and does not contain $\mathcal{M}$ as an ordered subgraph.
In fact, the best known upper bounds on $\overline{R}(\mathcal{G})$ for $n$-vertex ordered graphs $\mathcal{G}$ with bounded maximum degree and $\chi_<(\mathcal{G})=\chi$ are derived from the bound~\eqref{eq-degreeIntChrNumber} on $\overline{R}(\mathcal{G},\mathcal{K}^{(2)}_\chi(n))$.

The upper bound~\eqref{eq-diagonal} is quite close to the truth, as even when $\mathcal{H}$ is fixed we get a superexponential lower bound on $\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n))$. We note that since the first version of this preprint we learned that independently Fox and He (Theorem 1.3 in~\cite{foxhe19}) proved the same lower bound for the unordered Ramsey number and that implies the result of Theorem \ref{thm-3UnifMaxDegLower}. However we leave this result here as our proof is much simpler.

\begin{theorem}
\label{thm-3UnifMaxDegLower}
For every $t \geq 3$ and every positive integer $n$, we have
\[\overline{R}(\mathcal{K}^{(3)}_{t+1},\mathcal{K}^{(3)}_3(n)) \geq 2^{\Omega(n \log{n})}.\]
\end{theorem}

We do not have any nontrivial lower bound in the diagonal case $\overline{R}(\mathcal{H})$ for $\mathcal{H}$ with bounded maximum degree and $\chi_<(\mathcal{H})=3$.
We note that even for ordered graphs $\mathcal{G}$ with bounded maximum degree $d$ and $\chi_<(\mathcal{G})=2$ the best known lower bound on $\overline{R}(\mathcal{G})$ (and also on $\overline{R}(\mathcal{H})$) is only of order $\Omega((n/\log{n})^2)$~\cite{bjv16}, while the upper bound on $\overline{R}(\mathcal{G})$ is of order $n^{O(d)}$~\cite{bckk13,clfs17}.

Concerning $k$-uniform hypergraphs with $k>3$, the following result is based on a modification of the proof from~\cite[Proposition~6.3]{cfs10} and gives an estimate on ordered Ramsey numbers of ordered $k$-uniform hypergraphs with bounded interval chromatic number.
In particular, this estimate shows that we do not have a tower-type growth rate for $\overline{R}(\mathcal{H})$ once the uniformity and the interval chromatic number of $\mathcal{H}$ are bounded.

\begin{proposition}
\label{prop-generalBound}
Let $\chi,k$ be integers with $\chi \geq k \geq 2$ and let $\mathcal{H}$ be an ordered $k$-uniform hypergraph on $n$ vertices with interval chromatic number $\chi$.
Then there is a constant $c$ such that
\[\overline{R}(\mathcal{H}) \leq 2^{R^{\chi(\chi-1)}  (c\chi  n)^{\chi-1}},\]
where $R=R(K^{(k)}_\chi)$.
In particular, if the uniformity $k$ and the interval chromatic number $\chi$ of~$\mathcal{H}$ are fixed, we have
\[
\overline{R}(\mathcal{H}) \leq 2^{O(n^{\chi-1})}.
\]
\end{proposition}

Our understanding of the ordered Ramsey numbers of ordered hypergraphs is still very limited.
Many interesting open problem arose during our study and we would like to draw attention to some of them in Section~\ref{sec-openProblems}.

\section{Proof of Theorem~\ref{thm-3UnifMaxDegIntChr}}

Here we prove Theorem~\ref{thm-3UnifMaxDegIntChr} by giving an upper bound on the ordered Ramsey numbers of ordered 3-uniform hypergraphs with bounded maximum degree versus $\mathcal{K}^{(3)}_3(n)$.
We prove the following slightly stronger result.

\begin{theorem}
\label{thm-3UnifMaxDegIntChrPrecise}
Let $\mathcal{H}$ be an ordered 3-uniform hypergraph on $t$ vertices with maximum degree $d$, let $s$ be a positive integer and $\rho \in (0,1/8)$ be a real number.
Then there is a constant $C'$ such that
\[\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(s)) \leq t \cdot 2^{C'(s^{3/2} \cdot \rho^{-30d^2} \cdot d^6 +s^2\log{(\frac{1}{1-4\rho})})}.\]
\end{theorem}

First, we show that Theorem~\ref{thm-3UnifMaxDegIntChrPrecise} implies Theorem~\ref{thm-3UnifMaxDegIntChr}.

\begin{proof}[Proof of Theorem~\ref{thm-3UnifMaxDegIntChr}]
We can assume that $s > 8^{2+60d^2}$ by choosing the constant $C=C(d)$ from the statement of Theorem~\ref{thm-3UnifMaxDegIntChr} sufficiently large.
We then choose $\rho = s^{-1/(2+60d^2)}$, which gives $\rho < 1/8$ by our assumption on $s$.
Since $1-z \geq e^{-2z}$ for every $z$ with $0\leq z \leq 1/2$, we obtain $(1-4\rho)^{-s^2} \leq e^{8\rho s^2}$ and thus $s^2\log{(\frac{1}{1-4\rho})}) \leq 8s^2\rho$.
By our choice of~$\rho$, we then get
\[s^{3/2}\rho^{-30d^2} = s^2\rho = s^{2-1/(2+60d^2)}.\]
Therefore we can rewrite the upper bound from Theorem~\ref{thm-3UnifMaxDegIntChrPrecise} as \[\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(s)) \leq t \cdot 2^{C'(d^6 + 8)s^2\rho} \leq 2^{Cs^{2-1/(2+60d^2)}}\]
for sufficiently large $C=C(d)$,
which concludes the proof.
\end{proof}

In the rest of the section, we prove Theorem~\ref{thm-3UnifMaxDegIntChrPrecise}, but we first state some definitions.
For a graph $G$ and two disjoint subsets $X$ and $Y$ of vertices of $G$, we use $d(X,Y)$ to denote the \emph{edge density} between $X$ and $Y$, that is, $d(X,Y) = \frac{e(X,Y)}{|X||Y|}$, where $e(X,Y)$ denotes the number of edges with one vertex in $X$ and with the other one in $Y$.

For positive real numbers $\varepsilon_1,\varepsilon_2$, and $\rho$, we say that an ordered bipartite graph $\mathcal{G}$ with consecutive color classes $U$ and $V$ is \emph{bi-$(\varepsilon_1,\varepsilon_2,\rho)$-dense between $U$ and $V$}, if for all sets $X\subseteq U$ and $Y \subseteq V$ with $|X| \geq \varepsilon_1 |U|$ and $|Y| \geq \varepsilon_2 |V|$ we have $d(X,Y) \geq \rho$.

For positive real numbers $\varepsilon$ and $\rho$ and a positive integer $m$, an ordered 3-uniform hypergraph $\mathcal{H}$ is \emph{tri-$(\varepsilon,\rho,m)$-dense}, if for all consecutive subsets $V_1,V_2,V_3$ of vertices of $\mathcal{H}$, each of size at most $m$, and for all bipartite graphs $G_{1,2},G_{1,3},G_{2,3}$, each $G_{i,j}$ between $V_i$ and $V_j$, for which there are at least $\varepsilon m^3$ triangles with one edge in each $G_{i,j}$, at least $\rho$-proportion of these triangles forms hyperedges in $\mathcal{H}$.

The following embedding lemma is based on a similar result from~\cite{cfs12}.

\begin{lemma}\label{lem-embedding}
Let $\mathcal{H}$ be an ordered 3-uniform hypergraph on $t$ vertices with maximum degree $d$.
Let $ \varepsilon >0$ and $\rho \in (0,1)$ be two real numbers with $ \varepsilon \leq 2^{-6} \cdot \rho^{15d^2} \cdot d^{-3}$.
If $\mathcal{G}$ is a tri-$(\varepsilon,\rho,n/t)$-dense ordered 3-uniform hypergraph on $n \geq t/\varepsilon$ vertices, then $\mathcal{H}$ is an ordered subhypergraph of~$\mathcal{G}$.
\end{lemma}
\begin{proof}
Let $v_1 \prec \cdots \prec v_t$ be the vertices of $\mathcal{H}=(H,\prec)$.
For a vertex $v_j$ of $\mathcal{H}$ with $j>i$, we use ${\rm deg}^{{\preceq}i}_\mathcal{H}(j)$ to denote the number $\sum_{e \in E(\mathcal{H}): v_j \in e} |e \cap \{v_1,\dots,v_i\}|$.
Similarly, for vertices $v_j$ and $v_k$ of $\mathcal{H}$ and $i<j,k$, we use ${\rm deg}^{{\preceq}i}_\mathcal{H}(j,k)$ to denote the number of hyperedges of $\mathcal{H}$ containing $v_j$, $v_k$, and a vertex from $\{v_1,\dots,v_i\}$.
Observe that ${\rm deg}^{{\preceq}i}_\mathcal{H}(j) \leq 2d$  and ${\rm deg}^{{\preceq}i}_\mathcal{H}(j,k) \leq d$ for all $j$ and $k$ with $i<j,k$, as the maximum degree of $\mathcal{H}$ is $d$ and each hyperedge is multiplied by at most $2$.

We partition the vertex set of $\mathcal{G}$ into consecutive intervals $U_1,\dots,U_t$, each of size $n/t$.
We embed a copy $f(\mathcal{H})$ of $\mathcal{H}$ in $\mathcal{G}$ one vertex at time, embedding each vertex $f(v_i)$ of $f(\mathcal{H})$ to~$U_i$.
We proceed by induction on $i=0,1,\dots,t-1$.
Assuming that the vertices $v_1,\dots,v_i$ have been embedded as $f(v_1),\dots,f(v_i)$, we show that there are sets $U^i_{i+1},\dots,U^i_t$ and \allowbreak graphs $G^i_{j,k}$ with $i<j<k\leq t$ such that the following three conditions are satisfied:
\begin{enumerate}[label=(\roman*)]
    \item\label{item-embed1} $|U^i_j| \geq C^i_j \cdot n/t$, where $C^i_j = \rho^{d \cdot {\rm deg}^{{\preceq}i}_\mathcal{H}(j)}$,
    \item\label{item-embed2} $G^i_{j,k}$ is a bipartite graph between $U^i_j$ and $U^i_k$, which is bi-$(\varepsilon^i_j,\varepsilon^i_k,\rho^i_{j,k})$-dense between $U^i_j$ and $U^i_k$ with $\varepsilon^i_j = \rho^{4d^2 - d \cdot {\rm deg}^{{\preceq}i}_\mathcal{H}(j)}/(4d)$, $\varepsilon^i_k = \rho^{4d^2 - d \cdot {\rm deg}^{{\preceq}i}_\mathcal{H}(k)}/(4d)$, and $\rho^i_{j,k} = \rho^{{\rm deg}^{{\preceq}i}_\mathcal{H}(j,k)}$, and

    \item\label{item-embed3} for every $h \leq i$, every edge of $G^i_{j,k}$ forms a hyperedge of $\mathcal{G}$ with $f(v_h)$ if $\{v_h,v_j,v_k\}$ is a hyperedge of $\mathcal{H}$. Also, for all $h_1$ and $h_2$ with $h_1 < h_2 \leq i$, every vertex $u \in U^i_j$ with $i<j$ forms a hyperedge $\{ f(v_{h_1}),f(v_{h_2}),u\}$ of $\mathcal{G}$ if $\{v_{h_1},v_{h_2},v_j\}$ is a hyperedge of $\mathcal{H}$.
\end{enumerate}

For the induction base, assume $i=0$ and set $U^0_j = U_j$ for every $j \in \{1,\dots,t\}$.
For all $j$ and $k$ with $1 \leq j<k\leq t$, we let $G^i_{j,k}$ be the complete bipartite graph between $U^0_j$ and $U^0_k$.
Then the three conditions are trivially satisfied.

For the induction step, assume that the vertices $v_1,\dots,v_i$ have been embedded for some $i \geq 0$ while maintaining conditions~\ref{item-embed1}, \ref{item-embed2}, and \ref{item-embed3}.
We show how to embed the vertex $v_{i+1}$.

We let $W_{i+1}$ be the set of vertices $w$ from $U^i_{i+1}$ such that the neighborhood $U^i_j(w)$ of $w$ in $G^i_{i+1,j}$ has size at least $\rho^i_{i+1,j} |U^i_j|$ for every $j \in \{i+2,\dots,t\}$ such that $v_{i+1}$ and $v_j$ are contained in a hyperedge of $\mathcal{H}$.
Since the graph $G^i_{i+1,j}$ is bi-$(\varepsilon^i_{i+1},\varepsilon^i_j,\rho^i_{i+1,j})$-dense, there are at most $\varepsilon^i_{i+1}|U^i_{i+1}| \leq  \varepsilon^i_{i+1} n/t$ vertices in $U_{i+1}$ that have less than $\rho^i_{i+1,j} |U^i_j|$ neighbors in $|U^i_j|$.
Using condition~\ref{item-embed1} and the fact that the maximum degree of $\mathcal{H}$ is $d$, we see that the size of~$W_{i+1}$ satisfies
\begin{align*}
|W_{i+1}| &\geq |U^i_{i+1}| - 2d\varepsilon^i_{i+1}|U^i_{i+1}| \geq |U^i_{i+1}| - 2d\varepsilon^i_{i+1} n/t \geq C^i_{i+1} n/t - 2d\varepsilon^i_{i+1} n/t \geq C^i_{i+1} n/(2t),
\end{align*}
where the last inequality follows from $\varepsilon^i_{i+1} \leq C^i_{i+1}/(4d)$, as ${\rm deg}^{{\preceq}i}_\mathcal{H}(i+1) \leq 2d$.

For a vertex $w \in W_{i+1}$ and indices $j$ and $k$ such that $i+1<j<k$ and $\{v_{i+1},v_j,v_k\} \in E(\mathcal{H})$, we define the graph $H_{j,k}(w)$ as a subgraph of $G^i_{j,k}$ between $U^i_j(w)$ and $U^i_k(w)$ consisting of edges $\{x,y\}$ such that $\{w,x,y\} \in E(\mathcal{G})$.
We also let $W^{i+1}_{j,k}$ be the set of vertices $w \in W_{i+1}$ such that the graph $H_{j,k}(w)$ is not bi-$(\varepsilon^{i+1}_j,\varepsilon^{i+1}_k,\rho^{i+1}_{j,k})$-dense between $U^i_j(w)$ and $U^i_k(w)$.

By the definition of bi-$(\varepsilon^{i+1}_j,\varepsilon^{i+1}_k,\rho^{i+1}_{j,k})$-density, for every $w \in W^{i+1}_{j,k}$, there are sets $Y_j(w) \subseteq U^i_j(w)$ and $Y_k(w) \subseteq U^i_k(w)$ such that $|Y_j(w)| \geq \varepsilon^{i+1}_j|U^i_j(w)|$, $|Y_k(w)| \geq \varepsilon^{i+1}_k|U^i_k(w)|$, and $d(Y_j(w),Y_k(w)) < \rho^{i+1}_{j,k}$ in $H_{j,k}(w)$.
Since $w \in W_{i+1}$, we have
\[|Y_j(w)| \geq \varepsilon^{i+1}_j|U^i_j(w)| \geq \varepsilon^{i+1}_j\cdot \rho^i_{i+1,j} |U^i_j| \geq \varepsilon^i_j |U^i_j|,\]
where the last inequality follows from $\varepsilon^{i+1}_j\cdot \rho^i_{i+1,j} \geq \varepsilon^i_j$, as, since $\{v_{i+1},v_j,v_k\} \in E(\mathcal{H})$, we have ${\rm deg}_{\mathcal{H}}^{{\preceq}i+1}(j) > {\rm deg}_{\mathcal{H}}^{{\preceq}i}(j)$ and ${\rm deg}^{{\preceq}i}_\mathcal{H}(i+1,j) \leq d$.
Analogously, we obtain $|Y_k(w)| \geq \varepsilon^i_k |U^i_k|$.

We let $J_{i+1,j}$ be the graph connecting each $w \in W^{i+1}_{j,k}$ to vertices from $Y_j(w)$ and we analogously define the graph $J_{i+1,k}$.
The graph $G^i_{j,k}$ is bi-$(\varepsilon^i_j,\varepsilon^i_k,\rho^i_{j,k})$-dense between $U^i_j$ and $U^i_k$ by condition~\ref{item-embed2}.
Thus, since $|Y_j(w)| \geq \varepsilon^i_j |U^i_j|$ and $|Y_k(w)| \geq \varepsilon^i_k |U^i_k|$, the number of triangles in the tripartite graph between $W^{i+1}_{j,k}$, $U^i_j$, and $U^i_k$ formed by $J_{i+1,j}$, $J_{i+1,k}$, and $G^i_{j,k}$ is at least $\rho^i_{j,k}\sum_{w \in W^{i+1}_{j,k}}|Y_j(w)||Y_k(w)|$.

Suppose for contradiction that $|W^{i+1}_{j,k}| \geq |W_{i+1}|/(2d)$ for some $j$ and $k$ with $i+1<j<k$ and $\{v_{i+1},v_j,v_k\}\in E(\mathcal{H})$.
Then, since $|Y_j(w)| \geq \varepsilon^i_j |U^i_j|$ and $|Y_k(w)| \geq \varepsilon^i_k |U^i_k|$ for every $w \in W^{i+1}_{j,k}$,
\[\rho^i_{j,k}\sum_{w \in W^{i+1}_{j,k}}|Y_j(w)||Y_k(w)| \geq \rho^i_{j,k}\varepsilon^i_j\varepsilon^i_k |W^{i+1}_{j,k}||U^i_j||U^i_k| \geq \rho^i_{j,k}\varepsilon^i_j\varepsilon^i_k C^i_jC^i_k |W^{i+1}_{j,k}| (n/t)^2,\]
where the last inequality follows from condition~\ref{item-embed1}.
Using the assumption $|W^{i+1}_{j,k}| \geq |W_{i+1}|/(2d)$ and the fact $|W_{i+1}| \geq C^i_{i+1}n/(2t)$, we can estimate the above expression from below by
\[\rho^i_{j,k}\varepsilon^i_j\varepsilon^i_k C^i_jC^i_k C^i_{i+1} n^3/(4dt^3)\]
Our choice of parameters then gives $\rho^i_{j,k}\varepsilon^i_j\varepsilon^i_k C^i_jC^i_k C^i_{i+1}/(4dt^3) \geq \varepsilon/t^3$.
Altogether, there are at least
\[\rho^i_{j,k}\sum_{w \in W^{i+1}_{j,k}}|Y_j(w)||Y_k(w)| \geq \varepsilon (n/t)^3\]
triangles in the tripartite graph between $W^{i+1}_{j,k}$, $U^i_j$, and $U^i_k$ formed by $J_{i+1,j}$, $J_{i+1,k}$, and $G^i_{j,k}$.
Thus, since the ordered hypergraph $\mathcal{G}$ is tri-$(\varepsilon,\rho,n/t)$-dense, at least $\rho$-proportion of these triangles forms hyperedges in $\mathcal{G}$.
Therefore there are at least $\rho \cdot \rho^i_{j,k}\sum_{w \in W^{i+1}_{j,k}}|Y_j(w)||Y_k(w)|$ hyperedges of $\mathcal{G}$ between $W^{i+1}_{j,k}$, $U^i_j$, and $U^i_k$.

On the other hand, the number of hyperedges of $\mathcal{G}$ containing a vertex $w \in W^{i+1}_{j,k}$ and having an edge in each of the graphs $J_{i+1,j}$, $J_{i+1,k}$, and $G^i_{j,k}$ is the number of edges of~$H_{j,k}(w)$ between $Y_j(w)$ and $Y_k(w)$.
This number of edges is less than $\rho^{i+1}_{j,k}|Y_j(w)||Y_k(w)|$, as we know that $d(Y_j(w),Y_k(w)) < \rho^{i+1}_{j,k}$ in $H_{j,k}(w)$.
Thus the number of the hyperedges of $\mathcal{G}$ is less than
\[\rho^{i+1}_{j,k}\sum_{w \in W^{i+1}_{j,k}} |Y_j(w)||Y_k(w)| \leq \rho \cdot \rho^i_{j,k}\sum_{w \in W^{i+1}_{j,k}} |Y_j(w)||Y_k(w)|,\]
where the least inequality follows from $\rho^{i+1}_{j,k} \leq \rho \cdot \rho^i_{j,k}$, as $\{v_{i+1},v_j,v_k\} \in E(\mathcal{H})$ and thus we have ${\rm deg}_{\mathcal{H}}^{{\preceq}i+1}(j,k) > {\rm deg}_{\mathcal{H}}^{{\preceq}i}(j,k)$.
This contradicts the fact that there are at least $\rho~\cdot~ \rho^i_{j,k}\sum_{w \in W^{i+1}_{j,k}} |Y_j(w)||Y_k(w)|$ such hyperedges  of $\mathcal{G}$.

Thus $|W^{i+1}_{j,k}| < |W_{i+1}|/(2d)$.
In particular, the number of vertices $w \in W_{i+1}$ that do not lie in any set $W^{i+1}_{j,k}$ such that $i+1 < j <k \leq t$ and $\{v_{i+1},v_j,v_k\} \in E(\mathcal{H})$ is at least
\[|W_{i+1}| - \sum_{j<k: \{v_{i+1},v_j,v_k\} \in E(\mathcal{H})} |W^{i+1}_{j,k}| > |W_{i+1}| - \frac{d |W_{i+1}|}{2d} = \frac{|W_{i+1}|}{2},\]
as we are summing over at most $d$ pairs $(j,k)$, because the maximum degree of $\mathcal{H}$ is $d$.
Since $|W_{i+1}| \geq C^i_{i+1}n/(2t)$, we have at least $C^i_{i+1}n/(4t)$ such vertices and we let $f(v_{i+1})$ be any of them.
Since $n \geq t/\varepsilon > 4t/C^i_{i+1}$, at least one such vertex indeed exists.
For every $j \in \{i+2,\dots,t\}$ such that $v_{i+1}$ and $v_j$ are contained in a hyperedge of $\mathcal{H}$, we let $U^{i+1}_j=U^i_j(f(v_{i+1}))$.
We keep $U^{i+1}_j=U^i_j$ for all other values $j$.
Let $j$ and $k$ be indices such that $i+1 < j < k \leq t$.
If $\{v_{i+1},v_j,v_k\} \in E(\mathcal{H})$, we set $G^{i+1}_{j,k}=H_{j,k}(f(v_{i+1}))$.
For all other values of $j$ and $k$, we let $G^{i+1}_{j,k}$ be the subgraph of $G^i_{j,k}$ induced by $U^{i+1}_j$ and $U^{i+1}_k$.
In particular, if none of the vertices $v_j$ and $v_k$ lies in a hyperedge of $\mathcal{H}$ with $v_{i+1}$, we have $G^{i+1}_{j,k} = G^i_{j,k}$.

To finish the induction step, it remains to verify conditions~\ref{item-embed1}, \ref{item-embed2}, and~\ref{item-embed3}.
To verify condition~\ref{item-embed1}, first observe that if, for $j \in \{i+2,\dots,k\}$, the vertex $v_j$ is not in a hyperedge of $\mathcal{G}$ with $v_{i+1}$, then, by the choice of $C^i_j$ and $C^{i+1}_j$, $|U^{i+1}_j| = |U^i_j| \geq C^i_j n/t =C^{i+1}_j n/t$, as ${\rm deg}^{{\preceq}i}_\mathcal{H}(j) = {\rm deg}^{{\preceq}i+1}_\mathcal{H}(j)$.
Otherwise $|U^{i+1}_j| = |U^i_j(f(v_{i+1}))|$.
Since $f(v_{i+1}) \in W_{i+1}$, we have $|U^i_j(f(v_{i+1}))| \geq \rho^i_{i+1,j}|U^i_j|$.
Since ${\rm deg}^{{\preceq}i}_\mathcal{H}(j) < {\rm deg}^{{\preceq}i+1}_\mathcal{H}(j)$ and ${\rm deg}^{{\preceq}i}_\mathcal{H}(i+1,j) \leq d$, we have $\rho^i_{i+1,j}C^i_j \geq C^{i+1}_j$.
So we obtain $|U^{i+1}_j| \geq C^{i+1}_j n/t$.
Thus condition~\ref{item-embed1} is satisfied.

For condition~\ref{item-embed2}, let $j$ and $k$ be indices such that $i+1 < j<k\leq t$.
Consider first the case when $\{v_{i+1},v_j,v_k\} \in E(\mathcal{H})$.
Then $G^{i+1}_{j,k} = H_{j,k}(f(v_{i+1}))$.
Since $f(v_{i+1})$ does not lie in any set $W^{i+1}_{j,k}$ such that $i+1<j<k$ and $\{v_{i+1},v_j,v_k\} \in E(\mathcal{H})$, the graph $H_{j,k}(f(v_{i+1}))$ is bi-$(\varepsilon^{i+1}_j,\varepsilon^{i+1}_k,\rho^{i+1}_{j,k})$-dense between $U^i_j(f(v_{i+1}))$ and $U^i_k(f(v_{i+1}))$.
Thus $G^{i+1}_{j,k} = H_{j,k}(f(v_{i+1}))$ is bi-$(\varepsilon^{i+1}_j,\varepsilon^{i+1}_k,\rho^{i+1}_{j,k})$-dense between $U^{i+1}_j=U^i_j(f(v_{i+1}))$ and $U^{i+1}_k=U^i_k(f(v_{i+1}))$, which verifies condition~\ref{item-embed2} in this case.
Now, assume $\{v_{i+1},v_j,v_k\} \notin E(\mathcal{H})$.
If none of the two vertices $v_j$ and $v_k$ is in a hyperedge of $\mathcal{H}$ with $v_{i+1}$, then ${\rm deg}^{{\preceq}i+1}_\mathcal{H}(j) = {\rm deg}^{{\preceq}i}_\mathcal{H}(j)$, ${\rm deg}^{{\preceq}i+1}_\mathcal{H}(k) = {\rm deg}^{{\preceq}i}_\mathcal{H}(k)$, and ${\rm deg}^{{\preceq}i+1}_\mathcal{H}(j,k) = {\rm deg}^{{\preceq}i}_\mathcal{H}(j,k)$.
In particular, $\varepsilon^i_j = \varepsilon^{i+1}_j$, $\varepsilon^i_j = \varepsilon^{i+1}_j$, and $\rho^{i+1}_{j,k} = \rho^i_{j,k}$.
Since $U^{i+1}_j=U^i_j$, $U^{i+1}_k=U^i_k$, and $G^{i+1}_{j,k} = G^i_{j,k}$ is bi-$(\varepsilon^i_j,\varepsilon^i_k,\rho^i_{j,k})$-dense between $U^i_j$ and $U^i_k$, we see that $G^{i+1}_{j,k}$ is bi-$(\varepsilon^{i+1}_j,\varepsilon^{i+1}_k,\rho^{i+1}_{j,k})$-dense between $U^{i+1}_j$ and $U^{i+1}_k$, which again verifies condition~\ref{item-embed2}.
It remains to consider the case when exactly one of the vertices $v_j$ and $v_k$ is in a hyperedge of $\mathcal{H}$ with $v_{i+1}$.
By symmetry, we can assume without loss of generality that $v_j$ and $v_{i+1}$ are in a hyperedge of~$\mathcal{H}$.
Then  ${\rm deg}^{{\preceq}i+1}_\mathcal{H}(j) > {\rm deg}^{{\preceq}i}_\mathcal{H}(j)$, ${\rm deg}^{{\preceq}i+1}_\mathcal{H}(k) = {\rm deg}^{{\preceq}i}_\mathcal{H}(k)$, and ${\rm deg}^{{\preceq}i+1}_\mathcal{H}(j,k) = {\rm deg}^{{\preceq}i}_\mathcal{H}(j,k)$.
In particular, $\varepsilon^i_j > \varepsilon^{i+1}_j$, $\varepsilon^i_k = \varepsilon^{i+1}_k$, and $\rho^{i+1}_{j,k} = \rho^i_{j,k}$.
By definition, we have $U^{i+1}_j=U^i_j(f(v_{i+1}))$ and $U^{i+1}_k=U^i_k$.
Let $X$ be a subset of $U^{i+1}_j$ of size at least $\varepsilon^{i+1}_j|U^{i+1}_j|$ and let $Y$ be a subset of $U^{i+1}_k$ of size at least $\varepsilon^{i+1}_k|U^{i+1}_k| = \varepsilon^i_k|U^i_k|$.
We want to show that $d(X,Y) \geq \rho^{i+1}_{j,k}$ in $G^{i+1}_{j,k}$.
The subset $X$ has size at least $\varepsilon^i_j |U^i_j|$, as $|U^{i+1}_j| = |U^i_j(f(v_{i+1}))| \geq \rho^i_{i+1,j}|U^i_j|$ and our choice of $\varepsilon^{i+1}_j$ together with the fact ${\rm deg}^{{\preceq}i}_\mathcal{H}(i+1,j) \leq d$ gives $\varepsilon^{i+1}_j|U^{i+1}_j| \geq \varepsilon^i_j |U^i_j|$.
Thus, since $G^i_{j,k}$ is bi-$(\varepsilon^i_j,\varepsilon^i_k,\rho^i_{j,k})$-dense between $U^i_j \supseteq U^{i+1}_j$ and $U^i_k \supseteq U^{i+1}_k$, the density between $X$ and $Y$ is at least $\rho^i_{j,k}=\rho^{i+1}_{j,k}$ in $G^i_{j,k}$.
Since $G^{i+1}_{j,k}$ is an induced subgraph of $G^i_{j,k}$, the density between $X$ and $Y$ is also at least $\rho^i_{j,k}=\rho^{i+1}_{j,k}$ in $G^{i+1}_{j,k}$.
Thus $G^{i+1}_{j,k}$ is bi-$(\varepsilon^{i+1}_j,\varepsilon^{i+1}_k,\rho^{i+1}_{j,k})$-dense between $U^{i+1}_j$ and $U^{i+1}_k$, which verifies condition~\ref{item-embed2}.

We show that Condition~\ref{item-embed3} is satisfied as well.
Let $j$ and $k$ be indices such that $i+2<j<k\leq t$.
If $\{v_h,v_j,v_k\}$ is a hyperedge of $\mathcal{H}$ for some $h \leq i+1$, then $\{f(v_h),x,y\}$ is a hyperedge of $\mathcal{G}$ for every edge $\{x,y\}$ of $G^{i+1}_{j,k}$.
This is true for $h \leq i$ by the induction assumption, as $G^{i+1}_{j,k} \subseteq G^i_{j,k}$.
For $h=i+1$ we have $G^{i+1}_{j,k} = H_{j,k}(f(v_{i+1}))$ and the claim holds by the definition of $H_{j,k}(f(v_{i+1}))$.
Similarly, for all $h_1$ and $h_2$ with $h_1 < h_2 \leq i+1$, every vertex $u \in U^{i+1}_j$ with $i+1<j$ forms a hyperedge $\{f(v_{h_1}),f(v_{h_2}),u\}$ of $\mathcal{G}$ if $\{v_{h_1},v_{h_2},v_j\}$ is a hyperedge of $\mathcal{H}$.
This is because if $h_2\leq i$, then the claim follows from the induction assumption and the fact $U^{i+1}_j \subseteq U^i_j$.
For $h_2=i+1$, the triple $\{f(v_{h_1}),f(v_{h_2}),x\}$ is a hyperedge of $\mathcal{G}$ for every $x \in U^{i+1}_j = U^i_j(f(v_{h_2}))$ by the inductive assumption.

Finally, after we find all vertices $f(v_1),\dots,f(v_t)$, condition~\ref{item-embed3} ensures that they determine a copy of $\mathcal{H}$ as an ordered subhypergraph of $\mathcal{G}$.
\end{proof}

We use the following result proved by Conlon, Fox, and Sudakov~\cite{cfs12}.
It says that if a graph $G$ contains many triangles, a 3-uniform hypergraph whose hyperedges form a dense subset of the set of triangles in $G$ contains a large copy of $K^{(3)}_3(n)$.

\begin{lemma}[\cite{cfs12}]
\label{lem-tripartite}
Let $V_1,V_2,V_3$ be pairwise disjoint sets of vertices, each of size at most $m$, and let $G_{i,j}$ be a bipartite graph between $V_i$ and $V_j$ for all $i$ and $j$ with $1 \leq i < j \leq 3$.
Assume there are at least $\delta m^3$ triangles in the tripartite graph formed by $G_{1,2}$, $G_{1,3}$, and $G_{2,3}$.
Let $G$ be a $3$-uniform hypergraph containing at least $(1-\eta)$-proportion of the triangles in the tripartite graph, where $0<\eta < 1/8$.
Then $G$ contains a copy of $K^{(3)}_3(s)$ provided that
\[e^{2^{10}\delta^{-2}s^{3/2}}(1-4\eta)^{-4s^2}\left(\frac{\delta}{16}\right)^{4s} \leq m.\]
\end{lemma}

We note that, although it is not explicitly stated in the above lemma, the copy of $K^{(3)}_3(s)$ has $V_i$ as the $i$th color class.
Thus if the set $V_1 \cup V_2 \cup V_3$ is ordered and $V_1,V_2,V_3$ are consecutive in this ordering, we actually get a copy of the ordered hypergraph $\mathcal{K}^{(3)}_3(s)$.

We now proceed with the proof of Theorem~\ref{thm-3UnifMaxDegIntChrPrecise}, which implies Theorem~\ref{thm-3UnifMaxDegIntChr}.

\begin{proof}[Proof of Theorem~\ref{thm-3UnifMaxDegIntChrPrecise}]
Let $\mathcal{H}$ be an ordered 3-uniform hypergraph on $t$ vertices with maximum degree $d$, let $s$ be a positive integer and $\rho \in (0,1/8)$ be a real number.
We choose $\varepsilon = 2^{-6} \cdot \rho^{15d^2} \cdot d^{-3}$ and we let $N$ be an integer such that
\[N \geq t \cdot 2^{2^{28}\cdot s^{3/2} \cdot \rho^{-30d^2} \cdot d^6 +12s^2\log{(\frac{1}{1-4\rho})}}.\]
Note that $N \geq t/\varepsilon$.
Consider a red-blue coloring $\chi$ of the hyperedges of~$\mathcal{K}^{(3)}_N$.
We use $\mathcal{G}$ to denote the ordered $3$-uniform hypergraph on $N$ vertices formed by the hyperedges of~$\mathcal{K}^{(3)}_N$ that are blue in~$\chi$.
Similarly, we let $\overline{\mathcal{G}}$ be the hypergraph determined by red hyperedges of~$\mathcal{K}^{(3)}_N$ in~$\chi$.

If $\mathcal{G}$ is tri-$(\varepsilon,\rho,N/t)$-dense, then, since $N \geq t/\varepsilon$, Lemma~\ref{lem-embedding} implies that there is a blue copy of~$\mathcal{H}$ in $\chi$ and we are done.
Thus we assume that $\mathcal{G}$ is not tri-$(\varepsilon,\rho,N/t)$-dense.
That is, there are three consecutive subsets $V_1,V_2,V_3$ of vertices of $\mathcal{G}$, each of size at most $N/t$, and three bipartite graphs $G_{1,2},G_{1,3},G_{2,3}$, each $G_{i,j}$ between $V_i$ and $V_j$, for which there are at least $\varepsilon (N/t)^3$ triangles with one edge in each $G_{i,j}$ and less than $\rho$-proportion of these triangles forms hyperedges in $\mathcal{G}$.
By the choice of $\mathcal{G}$, at least $(1-\rho)$-proportion of these triangles forms hyperedges in $\overline{\mathcal{G}}$.

We show that \begin{equation}
\label{eq-tripartite}
e^{2^{10}\varepsilon^{-2}s^{3/2}}\cdot (1-4\rho)^{-4s^2}\cdot \left(\frac{16}{\varepsilon}\right)^{4s} \leq N/t.
\end{equation}
Then, by Lemma~\ref{lem-tripartite} applied with $\delta = \varepsilon$, $\eta = \rho$, and $m=N/t$, the ordered hypergraph $\overline{\mathcal{G}}$ contains a copy of $K^{(3)}_3(s)$ as an ordered subhypergraph.
By the definition of $\overline{\mathcal{G}}$, all hyperedges of such copy are red in $\chi$, which then finishes the proof.

To estimate~\eqref{eq-tripartite},
we estimate each of the three terms in the above expression separately by $N^{1/3}$.
The exponent in the first term in~\eqref{eq-tripartite} is
\[\log{(e)}2^{10}\varepsilon^{-2}s^{3/2} \leq 2^{23}\cdot \rho^{-30d^2}\cdot d^6 \cdot s^{3/2} \leq \frac{\log{(N/t)}}{3}.\]
The second term can be estimated by
\[(1-4\rho)^{-4 s^2} = 2^{-4s^2\log{(1-4\rho)}} \leq (N/t)^{1/3}.\]
Finally, the the third term satisfies
\[\left(\frac{16}{\varepsilon}\right)^{4s} = \left(2^{10}\cdot \rho^{-15d^2} \cdot d^3 \right)^{4s} < 2^{26\cdot s \cdot \rho^{-15d^2} \cdot d^3} \leq (N/t)^{1/3},\]
as $(2^{10}x)^4 < 2^{26x}$ for $x \geq 2$.
Altogether, both inequalities in~\eqref{eq-tripartite} are satisfied.
\end{proof}

\section{Proof of Theorem~\ref{thm-3UnifMaxDegLower}}

Here we prove a superexponential lower bound on $\overline{R}(\mathcal{K}^{(3)}_t,\mathcal{K}^{(3)}_3(n))$ for any $t \geq 4$.
First, we prove the following result, which gives superexponential lower bounds on ordered Ramsey numbers of 3-uniform hypergraphs $\mathcal{K}^{(3)}_{t+1}$ and $\mathcal{K}^{(3)}_3(n)$ provided that we have a superlinear lower bound on the ordered Ramsey number $\overline{R}(\mathcal{K}_t,\mathcal{K}_{m,m})$ in $m$.
The proof is inspired by the approach from~\cite{cfs10}.

\begin{lemma}
\label{lem-3UnifMaxDegLowerPrecise}
For $t \in \mathbb{N}$, if for $m = \lceil n/4\rceil$ we have $2n < \overline{R}(\mathcal{K}^{(3)}_t,\mathcal{K}_{m,m})^\alpha$ for some $\alpha \in (0,1]$,
then, for each sufficiently large $n$,
\[\overline{R}(\mathcal{K}^{(3)}_{t+1},\mathcal{K}^{(3)}_3(n)) \geq \left(\frac{(\overline{R}(\mathcal{K}_t,\mathcal{K}_{m,m})-1)^{1-\alpha}}{e^4}\right)^{(n+2)/6}.\]
\end{lemma}
\begin{proof}
We set $m = \lceil n/4 \rceil$ and $\ell = \lceil n/2 \rceil$.
Let $R$ denote the number $\overline{R}(\mathcal{K}_t,\mathcal{K}_{m,m})-1$ and let $N = (R^{1-\alpha}/e^4)^{(n+2)/6}$.
Using a probabilistic argument, we find a red-blue coloring $\chi$ of the hyperedges of~$\mathcal{K}^{(3)}_N=(K^{(3)}_N,<)$ that does not contain a blue copy of $\mathcal{K}^{(3)}_{t+1}$ and with positive probability does not contain a red copy of~$\mathcal{K}^{(3)}_3(n)$.

We use the following two auxiliary colorings.
Let $\chi_1$ be a red-blue coloring of the edges of~$\mathcal{K}_R$ that does not contain a blue copy of $\mathcal{K}_t$ nor a red copy of $\mathcal{K}_{m,m}$.
Such a coloring exists by the choice of $R$.
Let $\chi_2$ be a coloring of the edges of $\mathcal{K}_N$ with colors $1,\dots,R$, where the color of each edge is chosen uniformly independently at random from~$[R]$.
For three vertices $u < v < w$ of $\mathcal{K}^{(3)}_N$, we then set $\chi(u,v,w) = \chi_1(\chi_2(u,v),\chi_2(u,w))$ if $\chi_2(u,v)\neq \chi_2(u,w)$ and we let $\chi(u,v,w)$ be red otherwise.

Suppose for contradiction that $\chi$ contains a blue copy of $\mathcal{K}^{(3)}_{t+1}$ on some vertices $v_0, \dots, v_t$ of~$\mathcal{K}^{(3)}_N$.
Since all hyperedges of the copy of $\mathcal{K}^{(3)}_{t+1}$ are blue, our choice of $\chi$ gives distinct colors $\chi_2(v_0,v_1),\dots,\chi_2(v_0,v_t)$ and each edge $\{\chi_2(v_0,v_i),\chi_2(v_0,v_j)\}$ such that $\{v_0,v_i,v_j\}$ is a hyperedge of the blue copy of $\mathcal{K}^{(3)}_{t+1}$ is blue in $\chi_1$.
These blue edges determine a blue copy of~$\mathcal{K}_t$ in~$\chi_1$, which contradicts our choice of~$\chi_1$.

We now show that with positive probability there is no red copy of $\mathcal{K}^{(3)}_3(n)$ in $\chi$ by estimating the expected number of such copies.
Consider $3n$ vertices $v_1 < \cdots < v_{3n}$ of~$\mathcal{K}^{(3)}_N$ and suppose that these vertices induce a red copy of $\mathcal{K}^{(3)}_3(n)$ in $\chi$.
We fix $i \in [n]$.
Then there cannot be $2\ell$ distinct colors with $\ell$ of them among the colors $\chi_2(v_i,v_j)$ for $n<j \leq 2n$ and with the remaining $\ell$ of them among $\chi_2(v_i,v_k)$ for $2n < k \leq 3n$.
Otherwise such colors form a red copy of some ordering of $K_{\ell,\ell}$ in $\chi_1$ and every such ordering contains a copy of $\mathcal{K}_{m,m}$, as $\ell \geq 2m-1$.
Thus either all the colors $\chi_2(v_i,v_j)$ with $n < j \leq 2n$ are contained in the union of the set $\{\chi_2(v_i,v_k) \colon 2n<k \leq 3n\}$ together with a set of at most $\ell-1$ additional colors or, similarly, all the colors $\chi_2(v_i,v_k)$ with $2n < k \leq 3n$ are contained in the union of the set $\{\chi_2(v_i,v_j) \colon n<j \leq 2n\}$ together with a set of at most $\ell-1$ additional colors.

In the first case, all colors $\chi_2(v_i,v_j)$ with $n<j \leq 2n$ are contained in the union of the set $\{\chi_2(v_i,v_k) \colon 2n<k \leq 3n\}$ together with a set of at most $\ell-1$ additional colors.
In particular, the colors $\chi_2(v_i,v_j)$ with $n<j \leq 2n$ and $\chi_2(v_i,v_k)$ with $2n<k \leq 3n$ are all contained in a set of $n+\ell-1$ colors from $[R]$, as $|\{\chi_2(v_i,v_k) \colon 2n<k \leq 3n\}| \leq n$.
There are $\binom{R}{n+\ell-1}$ sets for the possible colors.
The probability that each of the colors $\chi_2(v_i,v_j)$ with $n<j \leq 2n$ and $\chi_2(v_i,v_k)$ with $2n<k \leq 3n$ is contained in a fixed set of $n+\ell-1$ colors from $[R]$ is $\left(\frac{n+\ell-1}{R}\right)^{2n}$.
The other case is symmetric.
Considering this for every $i \in [n]$, we see that the expected number of red copies of $\mathcal{K}^{(3)}_3(n)$ in~$\chi$ is at most
\begin{align*}
    &\binom{N}{3n}\left(2\binom{R}{n+\ell-1}^n\left(\frac{n+\ell-1}{R}\right)^{2n^2}\right) \\
    &\leq N^{3n}\left(2\left(\frac{eR}{n+\ell-1}\right)^{n(n+\ell-1)}\left(\frac{n+\ell-1}{R}\right)^{2n^2}\right)\\
    &\leq \left(N^3\left(2 e^{n+\ell-1}\left(\frac{n+\ell-1}{R}\right)^{n-\ell+1}\right) \right)^n \\
    &< \left(N^3e^{2n}\left(R^{\alpha-1}\right)^{n-\ell+1}\right)^n\\
    &\leq \left(N^3\left(e^4R^{\alpha-1}\right)^{n/2+1}\right)^n = 1,
\end{align*}
where we used $n+\ell-1\leq 2n \leq R^\alpha$ and our choice of $\ell$ and $N$.
Since the expected number of red copies of $\mathcal{K}^{(3)}_3(n)$ is less than $1$, there is a red-blue coloring $\chi$ that does not contain a blue copy of~$\mathcal{K}^{(3)}_{t+1}$ nor a red copy of~$\mathcal{K}^{(3)}_3(n)$.
\end{proof}

We also use the following bounds on Ramsey numbers of complete graphs and complete bipartite graphs proved by Li and Zang~\cite{liZan03}.

\begin{theorem}[\cite{liZan03}]
\label{thm-ramCliqueBipartite}
For every fixed integer $t \geq 3$ and every integer $m \ge 2$, there is a constant $c>0$ such that
\[R(K_t,K_{m,m}) \geq c\left(\frac{m}{\log{m}}\right)^{(t+1)/2}.\]
\end{theorem}

To finish the proof, we now combine all the auxiliary results.

\begin{proof}[Proof of Theorem~\ref{thm-3UnifMaxDegLower}]
It suffices to consider $\mathcal{K}^{(3)}_4$, as ordered Ramsey numbers do not decrease by adding vertices.
Let $n$ be a sufficiently large integer and let $m = \lceil n/4 \rceil$.
By Theorem~\ref{thm-ramCliqueBipartite}, we have $\overline{R}(\mathcal{K}_3,\mathcal{K}_{m,m}) \geq \Omega\left(\left(\frac{m}{\log{m}}\right)^2\right)$.
In particular, $2n < \overline{R}(\mathcal{K}_3,\mathcal{K}_{m,m})^\alpha$ for some fixed $\alpha < 1$ and a sufficiently large $n$.
Lemma~\ref{lem-3UnifMaxDegLowerPrecise} thus implies that
\[\overline{R}(\mathcal{K}^{(3)}_4,\mathcal{K}^{(3)}_3(n)) \geq \left(\frac{m}{\log{m}}\right)^{\Omega(n)} \geq 2^{\Omega(n\log{n})},\]
which finishes the proof.
\end{proof}

\section{Proof of Proposition~\ref{prop-generalBound}}

For integers $\chi$ and $k$ with $\chi \geq k \geq 2$ and an ordered $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with interval chromatic number $\chi$, we show that there is a constant $c$ such that
\[\overline{R}(\mathcal{H}) \leq 2^{R^{\chi(\chi-1)}  (c\chi  n)^{\chi-1}},\]
where $R=R(K^{(k)}_\chi)$.

Let $N=2^{ R^{\chi(\chi-1)}  (4\chi  n)^{\chi-1}}$ and consider a red-blue coloring of the hyperedges of $\mathcal{K}^{(k)}_N$ on $[N]$.
Since the interval chromatic number of $\mathcal{H}$ is $\chi$, we have $\mathcal{H} \subseteq \mathcal{K}^{(k)}_\chi(n)$ and thus it suffices to show that there is a monochromatic copy of $\mathcal{K}^{(k)}_\chi(n)$ in our coloring.

Since $R=R(K^{(k)}_\chi)$, every ordered subhypergraph of $\mathcal{K}^{(k)}_N$ induced by an $R$-tuple of vertices contains a monochromatic copy of $\mathcal{K}^{(k)}_\chi$ in our coloring.
Each such copy is contained in $\binom{N-\chi}{R-\chi}$ $R$-tuples of vertices of $\mathcal{K}^{(k)}_N$ and thus there are at least
\[\frac{\binom{N}{R}}{\binom{N-\chi}{R-\chi}} = \binom{R}{\chi}^{-1}\binom{N}{\chi}\]
monochromatic copies of $\mathcal{K}^{(k)}_\chi$.
Without loss of generality, we can assume that at least half of these copies has all hyperedges red.

Let $\mathcal{G}$ be the ordered $\chi$-uniform hypergraph with the vertex set $[N]$, where a $\chi$-tuple of vertices forms a hyperedge if and only if it induces a red copy of $\mathcal{K}^{(k)}_\chi$ in our coloring.
We know that $\mathcal{G}$ has at least $\frac{1}{2}\binom{R}{\chi}^{-1}\binom{N}{\chi} \geq \varepsilon \frac{N^\chi}{\chi!}$ hyperedges, where $\varepsilon = \frac{1}{4}\binom{R}{\chi}^{-1}$.
It is well-known (see~\cite{erdos64,niki09}, for example) that every $\chi$-uniform hypergraph on $N$ vertices with at least $\varepsilon \frac{N^\chi}{\chi!}$ hyperedges, where $(\ln{N})^{-1/(\chi-1)} \leq \varepsilon \leq \chi^{-3}$, contains a copy of $K^{(\chi)}_\chi(\lfloor \varepsilon(\ln{N})^{1/(\chi-1)} \rfloor)$.
Since \[\lfloor\varepsilon(\ln{N})^{1/(\chi-1)} \rfloor = \left\lfloor\frac{4R^\chi \chi n}{4\binom{R}{\chi}}\right\rfloor \geq \chi n,\]
we have a copy of $K^{(\chi)}_\chi(\chi n)$ in $\mathcal{G}$ in some ordering.
Note that every ordering of $K^{(\chi)}_\chi(\chi n)$ contains a copy of $\mathcal{K}^{(\chi)}_\chi(n)$ and thus $\mathcal{G}$ also contains a copy of $\mathcal{K}^{(\chi)}_\chi(n)$.
By the definition of $\mathcal{G}$, the vertices of this copy induce a red copy of $\mathcal{K}^{(k)}_\chi(n)$ in the coloring of~$\mathcal{K}^{(k)}_N$, which finishes the proof.
\qed

\section{Open problems}
\label{sec-openProblems}

There is a plenty of open questions about ordered Ramsey numbers for ordered hypergraphs, as this area is still vastly unexplored.
Here we offer some of the open problems that we considered during our study.

We proved estimates on the ordered Ramsey numbers of ordered $3$-uniform hypergraphs with bounded maximum degree and with interval chromatic number $3$.
However, our bounds are not tight.
Recall that it follows from Theorem~\ref{thm-3UnifMaxDegIntChr} and Theorem~\ref{thm-3UnifMaxDegLower} that there are positive constants $c_1$, $c_2$, and $\varepsilon$, all depending on $d$, such that every ordered $3$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with maximum degree $d$ satisfies
\[\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n)) \leq 2^{c_1n^{2-\varepsilon}},\]
while there is a fixed $\mathcal{G}$ such that
\[\overline{R}(\mathcal{G},\mathcal{K}^{(3)}_3(n)) \geq 2^{c_2n\log{n}}.\]
Thus although the exponents in the bounds are reasonably close in this non-diagonal case, there is still a gap between them and it would be interesting to close it.

\begin{problem}
Let $d$ be a fixed positive integer and let $\mathcal{H}$ be an ordered $3$-uniform hypergraph on $n$ vertices with maximum degree $d$.
Close the gap between the lower and upper bounds on $\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n))$.
\end{problem}

Another interesting problem is to extend the upper bound from Corollary~\ref{cor-3UnifMaxDegIntChr} to ordered $3$-uniform hypergraphs with bounded maximum degree and fixed interval chromatic number that is larger than $3$.
We recall that their ordered Ramsey numbers are bounded from above by $2^{O(n^2)}$ by Corollary~\ref{cor-3UnifBoundedIntChr}.
In particular, one can ask if the exponent is still subquadratic in the number of vertices.

\begin{problem}
Let $d$ and $\chi$ be fixed positive integers.
Is there an $\varepsilon = \varepsilon(d,\chi)>0$ such that, for every ordered $3$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with maximum degree $d$ and with interval chromatic number $\chi$, we have
\[\overline{R}(\mathcal{H}) \leq 2^{O(n^{2-\varepsilon})}?\]
\end{problem}

As we discussed in Subsection~\ref{subsec-previousResults}, the monotone hyperpaths are examples of ordered $3$-uniform hypergraphs with maximum degree $3$ and with an arbitrarily large interval chromatic number such that their ordered Ramsey numbers grow exponentially.
To our knowledge, this is the best lower bound on ordered Ramsey numbers of ordered $3$-uniform hypergraphs with bounded maximum degrees.
Can we obtain better lower bounds?

\begin{problem}
For a fixed $d$, is there an example of a family $\{\mathcal{H}_n\}$ of ordered $3$-uniform hypergraphs $\mathcal{H}_n$ on $n$ vertices with maximum degree $d$ such that the numbers $\overline{R}(\mathcal{H}_n)$ grow superexponentially in $n$?
\end{problem}

We note that even for $\mathcal{K}_n^{(3)}$ the best known lower bound on $\overline{R}(\mathcal{K}_n^{(3)})$ is of order $2^{\Omega(n^2)}$; see~\eqref{eq-hypergraphRamsey}.

The growth rate of Ramsey numbers of unordered $3$-uniform hypergraphs with bounded maximum degree is only linear in the number of vertices; see~\cite{cfs09}.
The situation is completely different for ordered $3$-uniform hypergraphs with bounded maximum degree, as we can, for example, see from the bounds for monotone hyperpaths.
In general, we are not aware of any nontrivial upper bounds on these ordered Ramsey numbers.

\begin{problem}
What is the upper bound on ordered Ramsey numbers of ordered $3$-uniform hypergraphs with bounded maximum degree?
\end{problem}

Finally, we proved all our upper bounds under the assumption that the maximum degree is bounded.
However, the corresponding bounds for ordered graphs such as~\eqref{eq-degreeIntChrNumber} hold for bounded degeneracy, which is a less restrictive assumption.
Thus one can also try to strengthen our results, in particular Theorem~\ref{thm-3UnifMaxDegIntChr} and Corollary~\ref{cor-3UnifMaxDegIntChr}, to ordered hypergraphs with bounded degeneracy instead of the maximum degree.

redpajama_set_name: RedPajamaArXiv


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text:
Nine Years And No New Ideas Sums Up Labour's Justice Policy
Tony Ryall
Nine years and no new ideas sums up Labour's Justice policy according to Justice Minister, Tony Ryall, today. "It's little wonder Labour chose to announce their policy on a Friday afternoon," said Mr Ryall.
"National on the other hand has brought new energy to law and order in New Zealand, building on a record of reducing crime, more police, and tougher penalties.
"During the passage of the Government's Home-Invasion legislation Labour's Justice Spokesman, Phil Goff, screamed at me from the Opposition benches that Labour supported tougher sentences for violent criminals. Labour's policy has no mention of tougher sentences.
"Then two weeks ago the Government introduced legislation to give victims new undeniable rights in the justice system. Suddenly the Opposition came out of their cryogenic freeze and realised victims were important. Up until then I don't recall a single mention of victims' rights from Mr Goff.
"Labour's new policy for victims follows the same line Mr Goff thought up in the House two weeks ago. Labour is saying the Government's Victims Rights Bill does not go far enough because it does not make the 'principles' contained in the old Act mandatory rather than discretionary.
"But it is not a question of simply changing "should" to "must" when it comes to principles of victims' rights. It is unclear which agency or agencies is/are responsible for complying with each of these principles. More work needs to be done on how to define responsibilities and enforce obligations to victims and their families under these principles.
"In April I directed the Ministry of Justice to begin this work and I expect a report in December. Further legislation can be expected late next year.
"However I was not prepared to delay the new statutory rights until this work is completed.
"Labour also talks of toughening bail conditions. The National-led Government introduced new bail laws on 1 June this year. These new laws will be far more effective at reducing offending while on bail than Labour's unworkable alternative.
"Like so much from this tired Opposition, the bits of their policy that are not copied directly from National, are ill-conceived, illogical or unnecessary," said Mr Ryall.

redpajama_set_name: RedPajamaCommonCrawl


Named for Dr. and Mrs. A. C. Gettys, and completed in 1965, this hall was the first residence hall built at UMHB since the completion of Burt and Stribling Halls in the 1920's. The two-story facility, which was remodeled in the summer of 2011, is one of two male residence halls on campus and houses 78 students in shared rooms. Gettys is located on the central quad area and is very close to Hardy Dining Hall and Mayborn Campus Center, which houses the gym, workout center, and recreational space. Gettys is also within easy walking distance to all of the academic buildings on campus, the intramural fields, and the library.
This facility includes a large lobby with game tables such as ping pong, and pool, a T.V. area, a computer lab, vending machines, and laundry rooms where laundry can be done free of charge. Internet and cable connections are located in each room. The common bathrooms located on each wing are cleaned daily by the housekeeping staff.
Approximate room size is 12' x 17', which is about 204 sq. ft.
Shared room has two twin beds with 80" mattresses, two desks, two desk chairs, and two chest of drawers. The chest of drawers are on casters and can also be moved under the beds. The room also has a built in vanity with a mirrored cabinet above the sink.
Beds are junior loft beds, which means they can be adjusted to create extra space beneath the bed. On the highest setting, the approximate space under the bed is 30", which allows the resident to place the chest of drawers under the bed. Bed lofts are also available to lease through Collegiate Bed Lofts. This is the only company approved for lease of bed lofts by the university. Students cannot build or bring in other bed lofts but can use bed risers, which come in different heights and can raise the bed anywhere from 3" to 8" inches. Collegiate Bed Loft often experiences high volume and availability of loft kits can become limited. It is important to lease your loft kit early.
Walk in closet is approximately 5' x 5'.
Community bathrooms, located on each wing of each floor, have multiple stalls and are equipped with showers.
Window measurements are 3' 4" x 5' 7" (inside casing) and mini-blinds are provided.
One microfridge is included in each room.
Floors are tile in a neutral tone and area rugs may be brought in.


Clinical question: Does the influenza vaccine lower the risk of major adverse cardiovascular events in adults with coronary disease?
Reference: Udell JA, Zawi R, Bhatt DL, et al. Association between influenza vaccination and cardiovascular outcomes in high-risk patients. A meta-analysis. JAMA 2013;310(16):1711-1720.
Synopsis: These investigators searched multiple sources including MEDLINE, EMBASE, the Cochrane Register, reference lists of eligible articles, clinicaltrials.gov, and conference abstracts without language restrictions for all published and unpublished randomized clinical trials comparing influenza vaccination with placebo or standard care. Two investigators independently reviewed potential studies for inclusion and methodologic quality using standard scoring tools. Disagreements were resolved by consensus. Six randomized controlled trials met inclusion criteria for the final meta-analysis. These trials (n = 6735) compared influenza vaccine with placebo or control for a mean duration of 7.9 months. The primary outcome measured was a composite of major adverse cardiovascular events, including cardiovascular death or hospitalization for myocardial infarction, unstable angina, stroke, heart failure, or urgent coronary bypass surgery. In the analysis of the 6 included trials, significantly fewer vaccinated patients developed a major adverse cardiovascular event compared with placebo or control patients (2.9% vs 4.7%, respectively; NNT = 58 95% CI, 38-124). The benefit of vaccination was strongest in the subset of patients with a history of recent acute coronary syndrome within the previous 6 months (10.25% with vaccine vs 23.1% with placebo or control; NNT = 8; 6-13). There was, however, no significant difference in all-cause mortality between the vaccinated and placebo or control patient groups. Formal statistical analyses found no evidence of significant heterogeneity among the trials or publication bias.
Clinical question: Are high-risk patients who are screened for diabetes better off than patients who are diagnosed though the usual means?
Reference: Simmons RK, Echouffo-Tcheugui JB, Sharp SJ, et al. Screening for type 2 diabetes and population mortality over 10 years (ADDITION-Cambridge): a cluster-randomised controlled trial. Lancet 2012;380(9855):1741-1748.
Synopsis: These British researchers conducted a cluster randomized trial of screening for type 2 diabetes. Fifteen general practices were to screen and intensively treat adults for diabetes, 13 practices were to screen and manage patients according to national guidelines, and 5 practices were left to their own devices. One practice dropped out before screening commenced. Overall, more than 20,000 high-risk adults, aged between 40 years and 69 years, were included and followed up for an average of 10 years. The authors used national databases to determine if any of the patients had died during the study period. Among the screening practices, 94% of potentially eligible patients were invited to be screened, 75% participated, and 3% were found to have diabetes. The overall death rate in the screened patients was 10.5 (per 1000 person-years) compared with 9.9 in unscreened patients. Additionally, there was no difference in cardiovascular deaths (3.3 and 3.2, respectively) or cancer-related deaths (4.8 and 4.4, respectively).
Clinical question: Is there any clinical benefit to a repeat bone mineral density screening test after an initial baseline screen in elderly men and women?
Reference: Berry SD, Samelson EJ, Pencina MJ, et al. Repeat bone mineral density screening and prediction of hip and major osteoporotic fracture. JAMA 2013;310(12):1256-1262.
Synopsis: These investigators analyzed data obtained from consenting adult participants of the ongoing Framingham cohort invited to have 3 BMD tests each approximately 4 years apart starting in 1987. Study participants (310 men and 492 women; mean age = 74.8 years) included those who had at least 2 BMD measures with a mean time between each of 3.7 years. Follow-up occurred for participants until death or through 2009 or 12 years after the second BMD. Individuals assessing medical records confirmed self-reported hip fractures, but no other major osteoporotic fractures, including spine, forearm, or shoulder. During a mean follow-up of 9.6 years, 1 or more major osteoporotic fracture occurred in 113 (14%) patients. Prediction modeling for hip or major osteoporotic fracture based on baseline BMD performed significantly better than models based on BMD change. Furthermore, adding BMD change as a variable to models using baseline BMD did not significantly improve prediction performance. Overall, the net change in the percentage of patients with a hip fracture reclassified with a second BMD as being at high risk was not significant (3.9%; 95% CI, -2.2% to 9.9%). Likewise, the net change in the percentage of patients without hip fracture reclassified as low risk by a second BMD was also not significant (-2.2%; 95% CI, -4.5% to 0.1%).
Clinical question: Are sigmoidoscopy and colonoscopy associated with a decreased likelihood of death due to colorectal cancer?
Reference: Nishihara R, Wu K, Lochhead P, et al. Long-term colorectal-cancer incidence and mortality after lower endoscopy. N Engl J Med 2013;369(12):1095-1105.
Synopsis: To date, there have been no randomized trials of screening colonoscopy. These researchers used observational data from the Nurse's Health Study and the Health Professionals Follow-up Study, 2 large prospective cohort trials that has followed women since 1976 and men since 1986, respectively. From 1988 through 2008, participants were asked whether they had undergone screening flexible sigmoidoscopy or colonoscopy. The authors reviewed the medical records and pathology reports of those who reported endoscopy to confirm the presence of polyps or CRC. The National Death Index was used to identify deaths due to any cause and deaths due to CRC. Persons undergoing colonoscopy were similar to those who chose not to have colonoscopy regarding the use of aspirin, multivitamins, alcohol, exercise, and red meat intake. Those who had no lower endoscopy were somewhat younger and were less likely to have a family history of CRC. The analysis adjusted for all of these variables when calculating multivariate hazard ratios for the primary outcomes. The likelihood of dying of CRC was lower among persons who had ever had flexible sigmoidoscopy (adjusted hazard ration [aHR] = 0.59; 95% CI, 0.45 - 0.76) and even lower among those who had ever undergone colonoscopy (aHR = 0.32; 0.24 - 0.45). Colonoscopy reduced the likelihood of death due to both proximal and distal cancers, whereas flexible sigmoidoscopy only reduced the likelihood of distal CRC mortality. The benefit was similar for men and women.
Clinical question: Does smoking cessation increase anxiety in smokers?
Reference: McDermott MS, Marteau TM, Hollands GJ, Hankins M, Aveyard P. Change in anxiety following successful and unsuccessful attempts at smoking cessation: cohort study. Br J Psychiatry 2013;202(1):62-67.
Synopsis: In this cohort study, researchers evaluated anxiety in 491 patients who received behavioral support, education, and nicotine replacement as aids to quit smoking. Patients who wished to quit were recruited from primary care practices. Approximately 18% of patients reported they smoked primarily "to cope," whereas approximately 22% reported smoking primarily "for pleasure," and the rest reported "both" as their motivations. Anxiety was measured using the Spielberger State-Trait Anxiety Inventory (STAI-6). Smoking status was determined based on self-report and biochemical validation. Sustained abstinence occurred in 14% of patients. Anxiety scores 6 months after starting smoking cessation therapy were 11.8 points lower in patients who quit smoking as compared with those who relapsed on a 60-point scale. At the start of the study, 14.7% of patients who eventually would quit were designated as having "high anxiety," and this percentage decreased to 10.3% after 6 months. In the patients who would relapse, 23.7% were highly anxious at the start and this percentage would increase to 31.0% at the end of the study (P = .009).


We are starting to push the incremental rollout of OxygenOS 9.0.3 for the OnePlus 6.
For those who were on previous versions for OnePlus 6, we appreciate your active feedback for reaching out to us. With your help, we have been able to better optimize and improve several key areas. Thank you for your patience and as always, we look forward to getting your continued feedback in the bug reporting forum here: https://forums.oneplus.com/feedback/.
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any news about OnePlus roaming?
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How's the battery life? Any improvements?
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text:
Angelica Arriola
Accountant VII • Department of Housing and Community Affairs
* Hire date refers to the current employer, the Department of Housing and Community Affairs. For employees with breaks in service, it is the most recent hire date.
Oct. 1, 2022 Accountant VII Department of Housing and Community Affairs 40 $88,172 +22%
July 1, 2022 Accountant VI Department of Housing and Community Affairs 40 $72,172 0%
April 1, 2022 Accountant VI Department of Housing and Community Affairs 40 $72,172 0%
Jan. 1, 2022 Accountant VI Department of Housing and Community Affairs 40 $72,172 0%
Sept. 30, 2021 Accountant VI Department of Housing and Community Affairs 40 $72,172
Accountant VII
Department of Housing and Community Affairs
Accountant VI
This Texas Tribune reporter covers the Department of Housing and Community Affairs. Send a tip via email or Twitter.


SIGNATURE has recently expanded throughout the PHILADELPHIA area and we have set the goal to expand ten-fold by the middle of this year!
We are a rapidly-growing promotional marketing company, applying a customer-friendly, direct, results-driven approach to our marketing research and sales. Because of our proven success and expertise, we continue to obtain new clientele as we dramatically increase our clients' product exposure, sales and brand recognition.
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Please see the blog post entitled "Radioactive Cutlery, Anyone?" for a full explanation of the purpose of this letter.
It was Dr. Gordon Edwards, President of the Canadian Coalition for Nuclear Responsibility, who drafted this letter.
I have sent a copy to our federal Environment Minister, Jim Prentice, & also to the other Canadian federal party leaders, under my own name (with the explanation that I support Dr. Edwards's concerns).
I have also sent it to my federal MP (Member of Parliament). If you don't know who your (Canadian) federal MP is, go here to find out her/his name & contact info.


text:
Israel: Soldiers Provoke Settler Violence; State Blocks UAE Oil Plan
July 26th, 2021 - by The New Arab Staff
New Report Details Israeli Military Complicity
In Settler Violence inside the West Bank
The New Arab Staff
(July 24, 2021) — A new report has documented the complicity of Israel's security forces in the violence perpetrated by Israeli settlers in the occupied West Bank.
The report by Breaking the Silence, which is run by Israeli veterans, said that Israeli forces were enabling the violence committed by Israeli settlers, and provided a "cloak of protection" for their actions, which were becoming increasingly more violent.
"Settlers, who are well aware of how to take advantage of this situation, often use the cloak of protection provided by the IDF to promote their goals, at the forefront of which is deepening their control of the area and pushing the Palestinians off of their land," the new report, titled On Duty, said.
The report complied the testimonies for 36 former members of the Israeli forces, dating back to 2012.
"These testimonies, like many others, describe a reality marked by unending harm to and threat toward Palestinian people, property and land," reads the report.
"The acts described here include different kinds of violent attacks, including beatings, stone-throwing and burning of agricultural fields and groves," it added.
Israeli forces have long been accused of complicity in the attacks committed by Israeli settlers against Palestinians, which have witnessed a recent drastic rise.
Israeli human rights organisation B'Tselem recorded that in just the first six months of 2021, attacks had increased 33% on the same period last year.
While the United Nations Office for the Coordination of Humanitarian Affairs said that more Palestinians had been injured by Jewish settlers in the first six months of 2021 than in the whole of 2020.
The testimonies in report highlight the differences in how Israeli forces operating in the West Bank dealt with settlers, compared with Palestinians.
Israel Freezes UAE Oil Deal over Environmental Concerns
The New Arab and Agencies
TEL AVIV (July 25, 2021) — Israel's environmental protection ministry said Sunday it was delaying implementation of a proposed oil transport deal with the United Arab Emirates, freezing a project that has angered environmentalists.
The agreement, which followed the UAE and Israel establishing diplomatic ties last year, would see Gulf oil brought to the Red Sea port of Eilat by tanker, than moved by pipeline through mainland Israel to the Mediterranean port of Ashkelon, from where it would be shipped to Europe.
The oil deal, involving Israel's state-owned Europe-Asia Pipeline Company (EAPC) and an Israeli-Emirati company called MED-RED Land Bridge Ltd, has not launched.
But activists have sounded the alarm about potential threats to the northern Red Sea corals off Eilat's coast.
Israeli environmental organisations challenged the plan in court, citing the risks of a devastating leak or spill, with tens of millions of tons of crude expected to be brought through Israel each year.
EAPC submitted its response in court last week, providing a risk assessment, which it claimed that the risk from the increased flow of crude was miniscule.
But on Sunday, Israel's environmental protection ministry said the risk assessment "did not meet the conditions" stipulated by the ministry, and was therefore not valid.
The ministry told EAPC in a letter it was "delaying the evaluation of your preparations to increase activity in the Eilat port, until the government has a discussion and reaches a decision" on the project.
The decision to freeze the deal's implementation was made by recently sworn-in Environmental Protection Minister Tamar Zandberg of the left-wing Meretz party, who has been an outspoken opponent of the EAPC-Emirati agreement.
A spokesman for the government of Prime Minister Naftali Bennett, whose ideologically diverse coalition was sworn in last month, said his office had "asked the court for an extension of time, in order to respond to the petition filed by the environmental organisations."
A spokeswoman for EAPC had no comment.
Activists argue the deal evaded tough regulatory scrutiny because of EAPC's status as a state-owned firm working in the sensitive energy sector.


text:
3 Kisses
Biography (current section)
2004 – present (17 years)
3 KISSES jumps from one high energy style to another with traces of just about every cool female-fronted rock band of the last 20 years. The band has been compared to an array of artists including the Cranberries, Luscious Jackson, the Breeders, Joan Jett, the Buzzcocks, Letters to Cleo, Wendy O. Williams, Blondie, the Ramones and everything in between. They have coined the phrase 'Texas Party Punk' to describe their music. 3 KISSES' live performance has been called 'freakin' explosive' and lead singer/guitarist Tish Meeks has been referred to as 'a ball of fire on stage.' 3 KISSES was launched in Austin, TX, in January 2004 and have since performed over 350 live shows for over 300,000 people. The band was featured on ABC's 'Wife Swap' in 2007 performing for almost 15 million viewers.
An idiosyncratic combination of alternative, pop, rock and punk filled with catchy guitar hooks and introspective lyrics, the heartfelt vocals and positive party vibe make the songs of 3 KISSES addicting. Tumultuous lyrics delivered in a zany, upbeat fashion has become a 3 KISSES trademark. Martin Folkman, Publisher of the Musician's Atlas writes, "3 KISSES is fronted by the kinetic Tish Meeks who no guitar, mic stand or stage can contain. packs more energy into a 3-minute song than all the WMD's combined!"
3 KISSES has been featured on the FOX 7 Morning Show in Austin, TX, WOAI-NBC in San Antonio, TX, KTRK-ABC in Houston, TX, KLBJ 93.7FM in Austin, KPFT 90.1 in Houston, Indie-Music.com, Female Musician Magazine, One Way Magazine, Origivation Magazine, The Macon Telegraph, The Brenham Banner Press, The College Station Eagle, EDGE Magazine, The Atlantic Monthly and The Austin American Statesman.
The band premiered their first music video Yellow in September 2004, released their debut CD Wings in November 2004, and released their sophomore album, It All Comes Down To Me in June 2006, which they followed up with an East Coast Tour. Their 3rd album, Nut Job was released in May 2007. Notable performances include multiple showcases during SXSW in Austin, Millennium Music Conference in Harrisburg, PA, Independent Music Conference in Philadelphia, PA, Dewey Beach Music Conference in Dewey Beach, DE and Austin's Heart of Texas Quadruple Bypass Festival.
In addition to their regular tour schedule, 3 KISSES supports breast cancer research by performing at Susan G. Komen Races across Texas. Tish's original story, Legacy, about the loss of her mother to breast cancer, appeared in Jack Canfield & Mark Victor Hansen's Chicken Soup for the Recovering Soul in December 2004.
3 KISSES is sponsored and/or endorsed by Paul Reed Smith Guitars, Daisy Rock Guitars, Dunlop, Miller Lite, Jägermeister, Coffin Case, Dirtbag Clothing, Old Dog Products, WB Custom Pickups, GoGirls Music, Independent Artist Registry, Texas Independent Musicians, Music Gorilla, Rockin' Moms and Powered by Sam Ash.
"3 Kisses completely own the audience with their punkish 'cuddle-angst'. Their songs are based on fun and intelligent mock anger–pierced tongue in cheek anthems to annoyance that everyone in the audience can relate to and smile about." - Dan Workman, SUGARHILL STUDIOS
"An exhilarating mix of pop tunes that keep you wanting more. Front woman Tish Meeks has a way of grabbing your attention and not letting it go." - Madalyn Sklar, founder of GoGirlsMusic.com
"One of the best compliments that I can give is that I have thousands of CDs stacked high but 3 Kisses stands alone in style." Eddie Russell, DJ
"3 Kisses is refreshing. Their sound is different, their lyrics are thoughtful and their music is addicting…a band to keep your eye on….they are going places." - Alexia Helfand, MUSIC GORILLA
"It's not often that you can pop something in and like every track, but 3 Kisses has got it!" Duss Rodgers, CATSASK MUSIC & ENTERTAINMENT
Version 4, edited by Tishker on 25 January 2010, 7:41pm
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Dried apricots soaked in orange juice, then wrapped in turkey bacon, make a tasty snack to pass around at a party. Take care when grilling the turkey bacon because it will dry out if overcooked. A tangy mustard mixture helps to add moisture during cooking.
Place the apricots in a small bowl, pour over the orange juice, and toss so that the apricots are moistened all over (this will prevent them from burning under the grill).
Mix the marmalade with the mustard. Spread each slice of turkey bacon with a little of the mustard mixture; then, using scissors, cut it in half lengthwise. Cut each piece in half again, this time across the middle, to make a total of 24 strips of turkey bacon.
Preheat a ridged grill pan or oven broiler to medium. Drain the apricots. Wrap a strip of turkey bacon around each apricot and secure it with a toothpick.
Arrange the turkey bites on the grill pan, then brush each with a little oil. Grill on each side until the turkey is just cooked, about 1 minute.
Pile the bites in a small, shallow bowl and sprinkle with chopped parsley, if using. Serve hot.
Each serving (two bites) provides: calories 58, calories from fat 18, fat 2 g, saturated fat 0 g, cholesterol 6 mg, carbohydrate 10 g, fibre 2 g, sugars 6 g, protein 2 g.