text:
\section{Introduction}
Ramsey Theory deals with the existence of some unavoidable structures as the number of vertices in a graph grows. In (classical) Ramsey numbers, we are interested in the minimum order of a graph which guarantees the existence of a clique or an independent set of given sizes.
It is well-known that the computation of Ramsey numbers is an extremely difficult task. One way to cope with this hardness is to consider Ramsey numbers in restricted graph classes. Ramsey numbers have been computed in planar graphs \cite{5}, in graphs with bounded degree \cite{6,7,8}, and in claw-free graphs \cite{9}. More recently, this approach has been also applied to perfect graphs and their subclasses in \cite{ramseygraphclassesHeggernes}.

Defective Ramsey numbers are a variation of (classical) Ramsey numbers where sparse and dense sets are used instead of independent sets and cliques. A \textit{$k$-sparse $j$-set} is a set $S$ of $j$ vertices of a graph $G$ such that each vertex in $S$ has at most $k$ neighbors in $S$. A \textit{$k$-dense $i$-set} is a set $D$ of $i$ vertices of a graph $G$ that is $k$-sparse in the complement of $G$; or alternatively, each vertex in $D$ misses at most $k$ vertices of $D$ in its neighborhood. A set is said to be \textit{$k$-defective} if it is either $k$-sparse or $k$-dense. The \textit{defective Ramsey number} $R_k^\mathcal{G}(i,j)$ is the smallest $ n $ such that all graphs on $n$ vertices in the class $\mathcal{G} $ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. A graph is said to be \textit{extremal} for  $R_k^\mathcal{G}(i,j)=n$ if it is a graph of order $n-1$ belonging to the class $\mathcal{G}$ and having neither $k$-dense $i$-set nor $k$-sparse $j$-set. If the defectiveness level $k=0$ (in which case the subscript 0 can be omitted), then it boils down to the classical Ramsey numbers. To establish that $R_k^\mathcal{G}(i,j)=n$ for some $n$, one should show two things: i) all graphs of order at least $n$ in $\mathcal{G}$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set; ii) there is at least one extremal graph for $R_k^\mathcal{G}(i,j)=n$. We note that besides being interesting for its own, finding all extremal graphs for a Ramsey number can be also helpful in establishing the next (for $i+1$ or $j+1$ where all the other parameters are the same) Ramsey numbers.

Defective Ramsey numbers have been first introduced in \cite{mynhardt} under the name of 1-dependent Ramsey numbers. Further defective Ramsey numbers have been computed in \cite{defectiveRamseyJohnChappell} and \cite{ekim1} using direct proof techniques. As expected, the approach of establishing new defective Ramsey numbers using direct proof techniques has quickly attained its limits. As a remedy, in \cite{defectiveparameterTinazAhu} and  \cite{defectiveRamseyJohnChappell}, computer-assisted efficient graph generation methods have been combined with direct proof techniques to obtain new defective Ramsey numbers (and related parameters). Besides efficient graph generation algorithms, a recent trend is to consider defective Ramsey numbers in graph classes. This has been initiated in \cite{1defectiveperfectTinazOylumJohn} where some 1-defective Ramsey numbers in perfect graphs have been computed. Further graph classes have been examined in \cite{defectiveramseynumbers} from the perspective of defective Ramsey numbers; namely, formulas for all or most cases have been established for defective Ramsey numbers in forests, cacti, bipartite graphs, split graphs and cographs, and the missing cases have been formulated as conjectures.

There is a close relationship between Ramsey numbers and the cocoloring problem where the vertices of a graph are partitioned into independent sets and cliques. A similar relationship exists between defective Ramsey numbers and the defective cocoloring problem where the vertices of a given graph are partitioned into $k$-defective sets. Given a graph $G$, a \textit{$k$-defective $m$-cocoloring} of $G$ consists in a partition of the vertex set of $G$ into $m$ subsets such that each one is a $k$-sparse or a $k$-dense set. The parameter $c_k(m)$ is defined as the maximum order $n$ such that every $n$-graph has a $k$-defective $m$-cocoloring. The classical version of this parameter (when $k=0$) has been studied by Straights in \cite{StraightsFormula} where $c_0(3)=8$ is shown. Further values such as $c_1(2)=7$, $c_1(3)\geq11$ and $c_2(2)\geq9$ are obtained by Ekim and Gimbel  \cite{ekim1}. This parameter has been formally defined by Akdemir and Ekim in \cite{defectiveparameterTinazAhu} where the values $c_0(4)=12$, $c_1(3)=12$ and $c_2(2)=10$ have been established via computer-assisted proofs using an efficient graph generation algorithm.

\textbf{Our contribution:}
In this paper, we first provide a generic algorithm to compute defective Ramsey numbers in various graph classes. In Section \ref{sec:algo}, we explain our algorithm which is based on the idea of constructing all graphs (starting from order one) that do not contain $k$-dense $i$-sets nor $k$-sparse $j$-sets, and that belong to the desired graph class. It takes as input the defectiveness level $k$ as well as the orders $i$ and $j$ for the $k$-dense and $k$-sparse sets, respectively. When used in a recursive way along with a recognition algorithm for the target graph class, it results in the value of the defective Ramsey number in that class and all extremal graphs for it. All of our codes and the extremal graphs we obtain are available at \href{https://github.com/yunusdemirci/DefectiveRamsey}{https://github.com/yunusdemirci/DefectiveRamsey} \cite{github}.

By combining our efficient graph generation algorithm with classical proof techniques, we extend the results in \cite{defectiveramseynumbers} and \cite{1defectiveperfectTinazOylumJohn} by establishing new defective Ramsey numbers in perfect graphs and bipartite graphs, and investigate chordal graphs which have not been addressed from this perspective to the best of our knowledge. In what follows, let $\mathcal{PG}$, $\mathcal{BIP}$ and $\mathcal{CH}$ denote the classes of perfect, bipartite and chordal graphs, respectively.

In Section \ref{sec:perfect}, we follow up the results in \cite{1defectiveperfectTinazOylumJohn} by establishing new defective Ramsey numbers in perfect graphs.  First, we give all extremal graphs for $R_1^{\mathcal{PG}}(4,8)$ (whose value has been already shown in \cite{1defectiveperfectTinazOylumJohn} without providing all extremal graphs), and then establish that $R_1^{\mathcal{PG}}(4,9)=19$ and $R_1^{\mathcal{PG}}(4,10)=22$. We also compute several $k$-defective Ramsey numbers in perfect graphs for $k\in\{2,3,4\}$ as reported in Tables \ref{table:2-defectiveperfect}, \ref{table:3-defectiveperfect} and \ref{table:4-defectiveperfect} respectively. To conclude, we conjecture that there exists $l\geq 3$ such that $R_1^{\mathcal{PG}}(4,j+l)\leq R_1^{\mathcal{PG}}(4,j)+3l$ for all $j\geq 1$.

Section \ref{sec:bip} is devoted to bipartite graphs. In \cite{defectiveramseynumbers}, all but exactly five 1-defective Ramsey numbers in bipartite graphs have been established and the conjecture $R_1^{\mathcal{BIP}}(4,j)=2j-1$ has been suggested for the five remaining values, namely for $j\in \{ 10,11,12,18,19\}$. In this paper, we show that this conjecture does not hold for $j=10$ and 11 by establishing that $R_1^{\mathcal{BIP}}(4,10)=18$ and $R_1^{\mathcal{BIP}}(4,11)=20$. Besides, we consider $k$-defective Ramsey numbers in bipartite graphs for $k\in\{2,3,4\}$ (for the first time) and provide several values in Tables \ref{table:2-defectivebipartite}, \ref{table:3-defectivebipartite} and \ref{table:4-defectivebipartite}.

In Section \ref{sec:chordal}, we consider $k$-defective Ramsey numbers in chordal graphs and report our findings in Tables \ref{table:1-defectivechordal}, \ref{table:2-defectivechordal}, \ref{table:3-defectivechordal} and  \ref{table:4-defectivechordal} for $k=1,2,3$ and 4 respectively. We also show that we have $R_k^{\mathcal{CH}}(i,k+2)=i$ for all $k\geq 1$ and $i\geq k+2$; $R_k^{\mathcal{CH}}(k+2,j)=j$ for all $k\geq 1$ and $j\geq k+2$; and $R_1^{\mathcal{CH}}(4,j)=2j-2$ for all $j\geq 3$ building up on a result on cactus graphs obtained in \cite{defectiveramseynumbers}.

It is important to note that the maximum defective Ramsey numbers we compute are of order 20 for both chordal and bipartite graphs, and of order 22 for perfect graphs. Indeed, it would not be possible to compute these values by considering all (unlabeled) graphs in a target graph class and checking for $k$-dense $i$-sets and $k$-sparse $j$-sets. The most efficient enumeration algorithms, by B. McKay \cite{McKay}, list all (unlabeled) connected chordal graphs up to 13 vertices, and all (unlabeled) perfect graphs up to 11 vertices. The efficiency of our algorithm is ensured by two main reasons. First, we do not generate all graphs in the target class and then check if each one admits the desired properties. Instead, we eliminate graphs which are not candidate for being extremal graphs at early stages of the generation. On top of that, we perform the recognition algorithms for graph classes efficiently by taking advantage of the fact that the new graphs are obtained by adding only one vertex to the previously obtained (sub-extremal) graphs which are already known to belong to the target graph class.

Besides computing new defective Ramsey numbers in perfect graphs, bipartite graphs and chordal graphs, we also initiate the study of the parameter $c_k(m)$ in various graph classes in Section \ref{sec:coco}. Formally, we define $c^{\mathcal G}_k(m)$ as the maximum order $n$ such that every $n$-graph in the graph class $\mathcal G$ has a $k$-defective $m$-cocoloring. We first establish a lower bound and an upper bound for the parameter $c^{\mathcal G}_k(m)$. Then, in Section \ref{sec:cocoperfect}, we focus on perfect graphs and show that $c_0^{\mathcal{PG}}(m)=m(m+3)/2$ for any $m$. Making use of efficient graph generation algorithms, we also establish two values, namely $c_1^{\mathcal{PG}}(2)=7$ (with exactly 24 extremal graphs) and $c_2^{\mathcal{PG}}(2)=11$, as well as the bounds $13 \leq c_1^{\mathcal{PG}}(3) \leq 14$ and $12 \leq c_3^{\mathcal{PG}}(2) \leq 14$. Lastly, in Section \ref{sec:cococograph}, we investigate the class of cographs, denoted by $\mathcal{CO}$, using solely direct proof techniques. The main result of this section is that $c_k^{\mathcal{CO}}(2)=3k+5$ for all $k\geq 1$.

\section{Computer Based Search for Extremal Graphs}\label{sec:algo}

In this section, we describe a generic algorithm to examine defective Ramsey numbers in a graph class. In subsequent sections, we will use this algorithm to derive some new values of defective Ramsey numbers in perfect graphs, bipartite graphs and chordal graphs, as well as to find related extremal graphs.  The codes of Algorithm \ref{algo:CHECK} for perfect graphs, bipartite graphs and chordal graphs and the lists of all extremal graphs obtained in this paper are available in our github account \cite{github}.

Let $\mathcal{T}_n^\mathcal{G}(k,i,j)$ be the set of all graphs of order $n$ belonging to the graph class $\mathcal{G}$ and containing no $k$-dense $i$-set or $k$-sparse $j$-set. For any integer $n$, we will call a $k$-dense $i$-set or a $k$-sparse $j$-set \textit{a forbidden $k$-defective set for} $\mathcal{T}_n^\mathcal{G}(k,i,j)$. Note that the set of all extremal graphs for ${R}_k^\mathcal{G}(i,j)$ corresponds to the set $\mathcal T_{n}^\mathcal{G}(k,i,j)$ for $n={R}_k^\mathcal{G}(i,j)-1$. Accordingly, a graph in  $\mathcal{T}_{n}^\mathcal{G}(k,i,j)$ for $n < {R}_k^\mathcal{G}(i,j)-1$ will be  called a \textit{sub-extremal} graph for ${R}_k^\mathcal{G}(i,j)$.

Our algorithm is based on the observation that every induced subgraph of an (sub-) extremal graph is a sub-extremal graph. Consequently, it starts with a set $W$ of graphs having no forbidden $k$-defective set for  $\mathcal{T}_n^\mathcal{G}(k,i,j)$ and produces the set $K$ of all graphs in $\mathcal{T}_{n+1}^\mathcal{G}(k,i,j)$ containing a graph in $W$ as induced subgraph. Clearly, if we start with $n=1$ and $W=\{K_1\}$, and run the algorithm recursively by taking the output set $K$ as input for the next run until the returned set $K$ is empty, then the last value of $n$ is equal to ${R}_k^\mathcal{G}(i,j)$ and the last non-empty output set $K$ is equal to the set of all extremal graphs for ${R}_k^\mathcal{G}(i,j)$. Alternatively, we can achieve the same goal by starting with the set of all sub-extremal graphs $\mathcal{T}_n^\mathcal{G}(k,i,j)$ for some $n<{R}_k^\mathcal{G}(i,j)$.\\

	\begin{algorithm}[H]

		\KwInput{$W\subseteq\mathcal{T}_n^\mathcal{G}(k,i,j)$, parameters $k, i, j$ such that $i,j\geq k+2$}
		\KwOutput{All graphs in $\mathcal{T}_{n+1}^\mathcal{G}(k,i,j)$ which contain at least one graph from $W$ as an induced subgraph}

		Let $K=\emptyset$. \\
		\For{$G\in W$}{
			\For{\label{line:S}$S\subseteq V(G)$}{
				Take the graph $G_S$ that is formed by adding a new vertex $v$ into $G$ and all edges between $v$ and all vertices in $S$.\\ \label{line:allS}
				Let $u=$ \textbf{TRUE}.\\ \label{line:candidate}
				\For{$I\subseteq V(G_S)$ such that $v\in I$ and $|I|\in\{i,j\}$}  {
					\If{$|I|=i$ and $G[I]$ is $k$-dense}{$u=$ \textbf{FALSE} and \textbf{BREAK}}
					\If{$|I|=j$ and $G[I]$ is $k$-sparse}{$u=$ \textbf{FALSE} and \textbf{BREAK}}
				} \label{line:kdef}

				 \If{u}{\label{line:class} \If{$G_S\in\mathcal{G}$}{\label{line:G}Add $G_S$ into $K$.}} \label{line:K}

		}}
		Return a maximal non-isomorphic set of graphs in $K$.

		\caption{Sub-extremal} \label{algo:CHECK}
	\end{algorithm}

	\vspace{0.3cm}

Technically, Algorithm \ref{algo:CHECK} takes as input a set $W$ of graphs on $n$ vertices which has no $k$-dense $i$-set and no $k$-sparse $j$-set. For each graph $G$ in the input set $W$, in lines \ref{line:S} and \ref{line:allS}, it constructs a new graph $G_S$ by adding a new vertex $v$ to $G$ in every possible way. Then, in lines \ref{line:candidate} to \ref{line:kdef}, it examines all subsets of the new graph $G_S$ containing $v$, and eliminates those containing a forbidden $k$-defective set. In lines \ref{line:class} to \ref{line:K}, only those graphs belonging to the graph class $\mathcal G$ among the remaining ones are added to the output set $K$. Finally, an isomorphism check taken from \cite{defectiveRamseyJohnChappell} is applied to return a maximal set of non-isomorphic graphs in $K$. At the end, Algorithm \ref{algo:CHECK} gives all graphs on $n+1$ vertices in the studied graph class which has no forbidden $k$-defective set and having at least one graph from the input as an induced subgraph.

In the following remark, we summarize how Algorithm \ref{algo:CHECK} can be used to produce all extremal graphs and thus the related defective Ramsey number for all parameters $k,i$ and $j$.

	\begin{remark}\label{rem:algo}
If we input $W=\mathcal{T}_n^\mathcal{G}(k,i,j)$, then Algorithm \ref{algo:CHECK} produces $\mathcal{T}_{n+1}^\mathcal{G}(k,i,j)$. For all integers $k,i$ and $j$, if we take $W=\mathcal{T}_1^\mathcal{G}(k,i,j)=\{K_1\}$ and apply Algorithm \ref{algo:CHECK} recursively until the set $K=\mathcal{T}_n^\mathcal{G}(k,i,j)$ is empty, then $R_k^{\mathcal G}(i,j)=n$ and $\mathcal{T}_{n-1}^\mathcal{G}(k,i,j)$ is the set of all extremal graphs for $R_k^{\mathcal G}(i,j)=n$.
	\end{remark}

All the algorithms in this paper are implemented in Python (except the improvement for the
lower bounds on $c_1^{\mathcal{PG}}(3)$ which is implemented in Julia) and executed on an Intel Core i7 machine with a 2.50-GHz clock speed and 8GB of RAM memory.

For the computation of 1-defective Ramsey numbers, we combined direct proof techniques with the use of Algorithm \ref{algo:CHECK} and set the time limit to two days for perfect graphs, one day for bipartite graphs and 12 hours for chordal graphs. For higher defectiveness values $k=2,3$ and 4, instead of setting a time limit, we rather limited the value of the defective Ramsey number to 15 in our tables. In most of the cases, these tables can be easily extended by allowing the algorithm to run longer; further details will be provided in related sections.

\section{Defective Ramsey Numbers in Perfect Graphs} \label{sec:perfect}

In this section, we use Remark \ref{rem:algo} to compute new defective Ramsey numbers in perfect graphs and determine their extremal graphs. In line \ref{line:G} of Algorithm \ref{algo:CHECK}, we use a straightforward algorithm for perfect graph recognition: according to the Strong Perfect Graph Theorem \cite{SPGT}, a graph $G$ is perfect if and only if neither $G$ nor its complement contains an \textit{odd hole}, that is, an induced cycle of odd length at least 5. To this end, we take all subsets of vertices of odd order at least five and eliminate the graph under consideration if such a set induces a cycle or the complement of a cycle. Note that we need to check perfectness only for graphs induced by subsets of vertices containing the newly added vertex since all the input graphs of Algorithm \ref{algo:CHECK} are known to be perfect by definition of $\mathcal{T}_n^\mathcal{PG}(k,i,j)$, thus they do not contain odd holes or their complements.

Our first result is related to $R_1^{\mathcal{PG}}(4,8)$; its value has been shown to be 15 in \cite{1defectiveperfectTinazOylumJohn}, however only one extremal graph (the Heawood graph depicted in Figure \ref{fig:IMAGEextremalfor4-1-8-1}) has been reported. Here, we show that there are exactly three extremal graphs.

	\begin{theorem}\label{thm:4-8-extremal-three}
	There are three extremal graphs for $R_1^{\mathcal{PG}}(4,8)=15$, shown in Figures \ref{fig:IMAGEextremalfor4-1-8-1}, \ref{fig:IMAGEextremalfor4-1-8-2}, and \ref{fig:IMAGEextremalfor4-1-8-3}.
	\end{theorem}
	\begin{figure}[h]
		\begin{minipage}{0.4\textwidth}

			\centering
			\includegraphics[scale=0.55]{extremal-1-4-8-1.png}
			\caption{Extremal graph $G_1$ for $R_1^{\mathcal{PG}}(4,8)$.} \label{fig:IMAGEextremalfor4-1-8-1}

		\end{minipage}
		\hfill
		\begin{minipage}{0.5\textwidth}

			\centering
			\includegraphics[scale=0.65]{extremal-1-4-8-2.JPG}
			\caption{Extremal graph $G_2$ for $R_1^{\mathcal{PG}}(4,8)$.} \label{fig:IMAGEextremalfor4-1-8-2}

		\end{minipage}

	\end{figure}

	\begin{figure}[h]

		\centering
		\includegraphics[scale=0.6]{extremal-1-4-8-3.JPG}
		\caption{Extremal graph $G_3$ for $R_1^{\mathcal{PG}}(4,8)$.} \label{fig:IMAGEextremalfor4-1-8-3}

	\end{figure}
\begin{proof}
By Remark \ref{rem:algo}, we identified $\mathcal{T}_{14}^\mathcal{PG}(1,4,8)$, that is all extremal graphs for $R_1^{\mathcal{PG}}(4,8)$, by running Algorithm \ref{algo:CHECK} with parameters $k=1$, $i=4$, $j=8$ and starting with $W=\mathcal{T}_{1}^\mathcal{PG}(1,4,8)=\{K_1\}$.
\end{proof}

	\noindent Our next result is obtained using a combination of direct proof techniques and computer assisted search. We show that $R_1^{\mathcal{PG}}(4,9)\leq 19$ by using theoretical approaches whereas the lower bound comes from the extremal graph found by Algorithm \ref{algo:CHECK}.

	\begin{theorem}\label{thm:4-9}
		We have $R_1^{\mathcal{PG}}(4,9)=19$ with the unique extremal graph $G_4$ given in Figure \ref{fig:extremalfor4-1-9}.
	\end{theorem}

	\begin{proof}
		Firstly, take a perfect graph $G$ on $19$ vertices. We will show that it has either a $1$-dense $4$-set or a $1$-sparse $9$-set. If it has a $1$-dense $4$-set, we are done. So assume $G$ has no $1$-dense $4$-set. This implies in particular that $N(v)$ is 1-sparse for every vertex $v\in V(G)$. We will prove that $G$ has a $1$-sparse $9$-set by examining two cases:
		\begin{itemize}
			\item[(i)] If there exists a vertex $v$ of degree at most three, then $G-N[v]$ is a perfect graph on at least 15 vertices. By Theorem \ref{thm:4-8-extremal-three}, $G-N[v]$ has a $1$-sparse $8$-set, say $S$. Then, $S\cup\{v\}$ is a $1$-sparse $9$-set in $G$, we are done.
			\item[(ii)] If all vertices have degree at least four, take an arbitrary vertex $v$, and let $\{v_1,v_2,v_3,v_4\}$ be four neighbors of $v$. Since $G$ has no $1$-dense $4$-set, we have $N(v_i)\cap N(v_j)=\{v\}$ for all $i\neq j$ and  $N(v_i)-N[v]$ is a $1$-sparse set with at least two vertices (since $d(v_i)\geq 4$) for all $i=1,2,3,4$. Moreover, there are no edges between $N(v_i)-N[v]$ and $N(v_j)-N[v]$ since $G$ has no induced cycle of length five (by perfectness) or four (in case $v_iv_j\in E$, by the absence of 1-dense 4-sets). As a result, $\{v\}\cup\big(N(v_1)-N[v]\big)\cup\big(N(v_2)-N[v]\big)\cup\big(N(v_3)-N[v]\big)\cup\big(N(v_4)-N[v]\big) $ contains a $1$-sparse $9$-set, we are done.
			\end{itemize}

Secondly, we show that there is only one extremal graph for $R_1^{\mathcal{PG}}(4,9)$ with 18 vertices. Let $H$ be a perfect graph on 18 vertices which has no $1$-dense $4$-set and no $1$-sparse $9$-set. Note that if all vertices of $H$ have degree at least four, then $H$ has a 1-dense 4-set or a 1-sparse 9-set using the same arguments as in case ii), a contradiction. If $H$ has a vertex $v$ of degree at most 2, then we obtain a 1-dense 4-set or a 1-sparse 9-set using the same arguments as in case i). It follows that $H$ has a vertex $v$ of degree three such that $G-N[v]\in \mathcal{T}_{14}^\mathcal{PG}(1,4,8)$. By Theorem \ref{thm:4-8-extremal-three}, we have $ \mathcal{T}_{14}^\mathcal{PG}(1,4,8)=\{G_1, G_2, G_3\}$. So, every extremal graph for $R_1^{\mathcal{PG}}(4,9)$ with 18 vertices should contain one graph in $\{G_1, G_2, G_3\}$. Therefore, $H$ can be  obtained by running Algorithm \ref{algo:CHECK} recursively by setting the parameters $k=1$, $i=4$, $j=9$ and starting with $W=\{G_1, G_2, G_3\}=\mathcal{T}_{14}^\mathcal{PG}(1,4,8)\subseteq \mathcal{T}_{14}^\mathcal{PG}(1,4,9)$ until output graphs have 18 vertices. It turns out that Algorithm \ref{algo:CHECK} returns only one extremal graph $G_4$ with 18 vertices which is depicted in Figure \ref{fig:extremalfor4-1-9}. In $G_4$, we note vertex 18 has degree three and $G- N[18]$ is isomorphic to $G_3$.
\begin{figure}[htbp]
		\centering
		\includegraphics[width=6cm,angle=0,height=6cm]{extremal-4-9-new.png}
		\caption{Unique extremal graph $G_4$ for $R_1^{\mathcal{PG}}(4,9)$.} \label{fig:extremalfor4-1-9}
	\end{figure}
\end{proof}

We now compute $R_1^{\mathcal{PG}}(4,10)$. Again, we construct an extremal graph by using our computer based approach, and then we combine both theoretical results and Algorithm \ref{algo:CHECK} to find the desired value. This time, we do not provide the full list of extremal graphs, but exhibit only one.

	\begin{theorem}\label{thm:4-10}
		We have $R_1^{\mathcal{PG}}(4,10)=22$.
	\end{theorem}

	\begin{proof}
		Firstly, we run Algorithm \ref{algo:CHECK} by setting the parameters $k=1$, $i=4$, $j=10$ and starting with $W=\mathcal{T}_{18}^\mathcal{PG}(1,4,9)=\{G_4\}\subseteq \mathcal{T}_{18}^\mathcal{PG}(1,4,10)$ recursively until the output is empty. We observe that the output is empty for $n=22$, meaning that there is no graph in $\mathcal{T}_{22}^\mathcal{PG}(1,4,10)$ which contains $G_4$ as an induced subgraph. On the other hand, we obtain a unique graph in $\mathcal{T}_{21}^\mathcal{PG}(1,4,10)$ that contains $G_4$ as an induced subgraph; the graph $G_5$ given in Figure \ref{fig:extremalfor1-4-10} has 21 vertices, $G_5-\{19,20,21\}$ induce $G_4$, and it has no $1$-dense $4$-set or $1$-sparse $10$-set. It follows that $R_1^{\mathcal{PG}}(4,10)\geq22$. We note that since we do not start with the complete set $\mathcal{T}_{18}^\mathcal{PG}(1,4,10)$ but instead with only a subset of it, our method do not guarantee the full list of extremal graphs for $R_1^{\mathcal{PG}}(4,10)$.

\begin{figure}[htbp]
	\centering
	\includegraphics[width=6cm,angle=0,height=6cm]{extremal-1-4-10.png}
	\caption{An extremal graph $G_5$ for $R_1^{\mathcal{PG}}(4,10)$.} \label{fig:extremalfor1-4-10}
\end{figure}

Now, let $G$ be a perfect graph on $22$ vertices. We will prove that $G$ has either a $1$-dense $4$-set or a $1$-sparse $10$-set. If $G$ has a 1-dense 4-set, we are done. So assume that $G$ has no $1$-dense $4$-set and we will prove that $G$ has a $1$-sparse $10$-set. As already stated, we know $G$ has no induced subgraph on $18$ vertices that is isomorphic to $G_4$, the unique graph in $\mathcal{T}_{18}^\mathcal{PG}(1,4,9)$. Thus, every induced subgraph of $G$ on $18$ vertices has a $1$-sparse $9$-set. Now, take a vertex $v\in V(G)$. If $d(v)\leq 3$, then $G-N[v]$ has at least $18$ vertices, thus contains a $1$-sparse $9$-set, say $J$. Then $J\cup\{v\}$ is a $1$-sparse $10$-set and we are done. Therefore, we assume that all degrees in $G$ are at least four.
		\begin{itemize}
			\item[(i)] If there exists a vertex of degree at least five, say $v$, let $v_1$,$v_2$,$v_3$,$v_4$,$v_5$ be five neighbors of $v$. Since $G$ has no $1$-dense $4$-set, $\{v_1,v_2,v_3,v_4,v_5\}$ is $1$-sparse and any two of these five vertices have a unique common neighbor which is $v$. Moreover, for any $i\neq j$, we have that no neigbor of $v_i$ is adjacent to a neighbor of $v_j$ because $G$ has no induced cycle of lenght 5 (or 4). Since $d(v_i)\geq 4$ for all $i\in\{1,2,3,4,5\}$, the set $\bigcup_{i=1}^{5} \big(N(v_i)\backslash\{v\}\big)-\{v_1,v_2,v_3,v_4,v_5\}$ contains a $1$-sparse $10$-set, we are done.
			\item[(ii)] If all vertices have degree four, take a vertex $v$ and its four neighbors $v_1$,$v_2$,$v_3$,$v_4$. Since ${S}=\{v,v_1,v_2,v_3,v_4\}$ has no $1$-dense $4$-set and $d(v_i)=4$ for all $i\in\{1,2,3,4\}$, $N(v_i)-{S}$ has at least two vertices for all $i$. If $|N(v_i)-{S}|\geq 3$ for some $i$, then $\bigcup_{i=1}^{4} N(v_i)-N(v)$ has a $1$-sparse $10$-set since $G$ has no $1$-dense $4$-set and no cycle of lenght $5$, we are done. So, assume $|N(v_i)-{S}|=2$ for all $i$, thus there are exactly two edges with both endpoints in $N(v)$, say without loss of generality $v_1v_2, v_3v_4\in E(G)$. Note that $v$ is selected arbitrarily, therefore we can suppose that $N(u)$ contains exactly two edges for all $u\in G$. Let $N(v_i)-\mathcal{S}=\{a_i,b_i\}$ for all $i$. Observe that $a_ib_i\in E(G)$ for all $i$ by our assumption. Moreover, since $G$ has no $1$-dense $4$-set and no cycle of size $5$, there are no edges between two sets $\{a_i,b_i\}$ and $\{a_j,b_j\}$ for all $i\neq j$. Consider the sets $U=\big(N(a_1)\cup N(b_1)\big)-\{v_1,a_1,b_1\}$ and $V=\big(N(a_2)\cup N(b_2)\big)-\{v_2,a_2,b_2\}$. Observe that $|U|=|V|=4$ and they are disjoint. Since $G$ has exactly 22 vertices, there exists a unique vertex in $V(G)-({S}\cup\{a_1,b_1,a_2,b_2,a_3,b_3,a_4,b_4\}\cup U\cup V)$, say $p$. Note that $p$ has no neighbors in ${S}\cup\{a_1,b_1,a_2,b_2\}$ because $G$ has no $1$-dense $4$-set and no cycle of size $5$. Since $d(p)=4$, by the pigeonhole principle, $p$ has at least two neighbors from one of the sets $\{a_3,b_3,a_4,b_4\}$, $U$ and $V$, which yields a $1$-dense $4$-set or a 5-cycle, a contradiction.
		\end{itemize}
\end{proof}

We computed additional defective Ramsey numbers in perfect graphs by using Algorithm \ref{algo:CHECK}. Our results for defectiveness levels $k=1,2,3$ and 4 are reported in Tables \ref{table:1-defectiveperfect}, \ref{table:2-defectiveperfect}, \ref{table:3-defectiveperfect} and \ref{table:4-defectiveperfect}, respectively, with number of extremal graphs in parentheses (whenever we could obtain their full list).

The values found in this study are highlighted in bold font. Values which are already known in Table \ref{table:1-defectiveperfect} are obtained in \cite{1defectiveperfectTinazOylumJohn}. For the remaining tables, the only values which are already known follow from Lemma 2.1 in \cite{defectiveramseynumbers} which states that $R_k^{\mathcal{PG}}(k+2,j) =j$ for all $j \geq k + 2$ (also $R_k^{\mathcal{PG}}(i,k+2)=i$ for all $i\geq k+2$, by self-complementarity of perfect graphs) where $k\in\{2,3,4\}$. While we execute our algorithm for two days when we deal with the values in the $1$-defective case, we could obtain all the values up to 15 in less than $4$ hours for $k$-defective cases for $k\geq 2$. In all computations, isomorphism checks are the most time consuming part as compared to checking perfectness and forbidden $k$-defective sets.\\

\begin{table}[H]\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_1^{\mathcal{PG}}(i,j)$} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10} \\
					\hline

					\textbf{3} & 3\textbf{(2)} & 4\textbf{(2)} & 5\textbf{(3)} & 6\textbf{(3)} & 7\textbf{(4)} & 8\textbf{(4)} & 9\textbf{(5)} & 10\textbf{(5)} \\
					\hline

					\textbf{4} & 4\textbf{(2)} & 6(1) & 8(2) & 10(4) & 13(3) & 15\textbf{(3)} & \textbf{19(1)} & \textbf{22}\\
					\hline

					\textbf{5} & 5\textbf{(3)} & 8(2) & 13\textbf{(2)} &  &  & &&\\
					\hline

				\end{tabular}
			}

			\caption{1-Defective Ramsey Numbers in Perfect Graphs}
			\label{table:1-defectiveperfect}

		\end{table}
		\begin{table}[H]

			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_2^{\mathcal{PG}}(i,j)$} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10}\\
					\hline

					\textbf{4} & 4\textbf{(4)} & 5\textbf{(4)} & 6\textbf{(4)} & 7\textbf{(4)} & 8\textbf{(4)} & 9\textbf{(4)} &10\textbf{(4)} \\
					\hline

					\textbf{5} & 5\textbf{(4)} & \textbf{7(2)} & \textbf{8(13)} & \textbf{10(16)} & \textbf{12(6)} & \textbf{15(2)}&\\
					\hline

					\textbf{6} & 6\textbf{(4)} & \textbf{8(13)} & \textbf{10(2)} & \textbf{13(7)} & & &\\
					\hline

				\end{tabular}
			}

			\caption{2-Defective Ramsey Numbers in Perfect Graphs}
			\label{table:2-defectiveperfect}

		\end{table}
		\begin{table}[H]

			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_3^{\mathcal{PG}}(i,j)$} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10}& \textbf{11} \\
					\hline

					\textbf{5} & 5\textbf{(11)} & 6\textbf{(11)} & 7\textbf{(12)} & 8\textbf{(12)} & 9\textbf{(13)} & 10\textbf{(13)}& 11\textbf{(14)}\\
					\hline

					\textbf{6} & 6\textbf{(11)} & \textbf{8(4)} & \textbf{9(28)} & \textbf{10(159)} & \textbf{12(3)} & \textbf{13(67)} & \\
					\hline

					\textbf{7} & 7\textbf{(12)} & \textbf{9(28)} & \textbf{11(4)} & & & &\\
					\hline

				\end{tabular}
			}

			\caption{3-Defective Ramsey Numbers in Perfect Graphs}
			\label{table:3-defectiveperfect}

		\end{table}
		\begin{table}[H]

			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_4^{\mathcal{PG}}(i,j)$} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10}& \textbf{11} & \textbf{12} \\
					\hline

					\textbf{6} & 6\textbf{(33)} & 7\textbf{(33)} & 8\textbf{(33)} & 9\textbf{(33)} & 10\textbf{(33)} & 11\textbf{(33)} & 12\textbf{(33)} \\
					\hline

					\textbf{7} & 7\textbf{(33)} & \textbf{9(11)} & \textbf{10(84)} & \textbf{11(549)} & \textbf{13(4)}& \textbf{14(28)} &\\
					\hline

					\textbf{8} & 8\textbf{(33)} & \textbf{10(84)} & \textbf{12(8)} & & & &\\
					\hline

				\end{tabular}
			}

			\caption{4-Defective Ramsey Numbers in Perfect Graphs}
			\label{table:4-defectiveperfect}
		\end{table}

We conclude this section with a conjecture on the growth of $R_1^{\mathcal{PG}}(4,j+3)$ for $j\geq 1$. A formula for the classical Ramsey numbers in perfect graphs has been provided in \cite{ramseygraphclassesHeggernes}. Namely, $R^{\mathcal{PG}}(i, j) = (i-1)(j-1) + 1$ for all $i, j \geq 1$. It follows that $R_0^{\mathcal{PG}}(4,j)=3j-2$. Since $R_1^{\mathcal{PG}}(4,j)\leq R_0^{\mathcal{PG}}(4,j)$, the value of $R_1^{\mathcal{PG}}(4,j)$ does not increase by more than three for consecutive vales of $j$ in the long run. However, the inequality $R_1^{\mathcal{PG}}(4,j+1)\leq R_1^{\mathcal{PG}}(4,j) +3$ does not hold in the light of our results $R_1^{\mathcal{PG}}(4,8)=15$ and $R_1^{\mathcal{PG}}(4,9)=19$. Likewise, $R_1^{\mathcal{PG}}(4,j+2)\leq R_1^{\mathcal{PG}}(4,j) +6$ does not hold neither since $R_1^{\mathcal{PG}}(4,8)=15$ and $R_1^{\mathcal{PG}}(4,10)=22$. Instead, we expect that the difference can be at most $3l$ in each $l$ steps for some fixed $l\geq 3$, which holds for currently known values.  Accordingly, we formulate the following conjecture.

	\begin{conjecture}
		There exists $l\geq 3$ such that $R_1^{\mathcal{PG}}(4,j+l)\leq R_1^{\mathcal{PG}}(4,j)+3l$ for all $j\geq 1$.
	\end{conjecture}

	\section{Defective Ramsey Numbers in Bipartite Graphs} \label{sec:bip}
	\noindent
In \cite{defectiveramseynumbers}, it has been shown that $R_1^{\mathcal{BIP}}(4,j)=2j-1$ for all $j\geq 8$ except for $j\in \{ 10,11,12,18,19\}$. It has been also conjectured that for $j\in \{ 10,11,12,18,19\}$, we also have $R_1^{\mathcal{BIP}}(4,j)=2j-1$. In this section, we first establish two of these missing numbers, namely for $j=10$ and $j=11$  by combining classical proof techniques and Algorithm \ref{algo:CHECK}. It turns out that the conjecture reflecting the general trend does not hold for $j=10$ and $j=11$. Subsequently, we compute some new values of $k$-defective Ramsey numbers in bipartite graphs for $k\in \{2,3,4\}$, also by using Algorithm \ref{algo:CHECK}.

	\begin{theorem}
		$R_1^{\mathcal{BIP}}(4,10)=18$.
	\end{theorem}

	\begin{proof}
		From  \cite{defectiveramseynumbers}, we know $R_1^{\mathcal{BIP}}(4,9)=17$ which trivially implies $R_1^{\mathcal{BIP}}(4,10)\geq18$ since we can add an isolated vertex to the (unique) extremal graph of $R_1^{\mathcal{BIP}}(4,9)$ (which has 16 vertices) and obtain a bipartite graph which has no $1$-dense $4$-set and no $1$-sparse $10$-set. Now, we will show that any bipartite graph on $18$ vertices has either a $1$-dense $4$-set or a $1$-sparse $10$-set. Take a bipartite graph $G$ on $18$ vertices. If $G$ has a $1$-dense $4$-set, we are done, thus assume that $G$ has no $1$-dense $4$-set. We will prove that $G$ has a $1$-sparse $10$-set. Suppose there exists a vertex $v$ of degree at most $3$.  If $G-N[v]$ has a $1$-sparse $9$-set, say $J$, then $J\cup \{v\}$ is a $1$-sparse $10$-set. Otherwise, $G-N[v]\in\mathcal{T}_{m}^\mathcal{BIP}(1,4,9)$ for some $m\geq 14$. Noting that all graphs in $\mathcal{T}_{m}^\mathcal{BIP}(1,4,9)$ where $m\geq 14$ can be produced by the Algorithm \ref{algo:CHECK} by setting the parameters $k=1$, $i=4$, $j=10$ and starting with $W=\mathcal{T}_{14}^\mathcal{BIP}(1,4,9)\subseteq\mathcal{T}_{14}^\mathcal{BIP}(1,4,10)$. We run Algorithm \ref{algo:CHECK} with these settings and observe that there is no bipartite graph on $18$ vertices that has an induced subgraph from $\mathcal{T}_{m}^\mathcal{BIP}(1,4,9)$ for $m\geq14$. So, $G-N[v]$ has again a $1$-sparse $9$-set, say $J$, then $J\cup \{v\}$ is a $1$-sparse $10$-set and we are done.

Now, assume that all vertices in $G$ have degree at least four. Take a vertex $u$ with its four neighbors $u_1$,$u_2$,$u_3$,$u_4$. Since $G$ has no $1$-dense $4$-set, $u$ is the unique common neighbor of any two of $x_1$, $x_2$, $x_3$, $x_4$. Moreover, since $G$ is bipartite, $N(x_1)\cup N(x_2)\cup N(x_3)\cup N(x_4)$ is an independent set (thus a 1-sparse set) of size at least $13$, which completes the proof.
	\end{proof}

\begin{theorem}\label{thm:bipartite_1_4_11}
	$R_1^{\mathcal{BIP}}(4,11)=20$.
\end{theorem}

\begin{proof}
	From \cite{defectiveramseynumbers}, we know that there is a unique extremal graph, say $H$, on 16 vertices for $R_1^{\mathcal{BIP}}(4,9)=17$. Consider the disjoint union of $H$ and $K_{1,2}$. Clearly, this graph is a bipartite graph on 19 vertices and it has no $1$-dense $4$-set and no 1-sparse 11-set. Thus, $R_1^{\mathcal{BIP}}(4,11)\geq 20$.

Now, let $G$ be a bipartite graph on 20 vertices and let us show that it has either a 1-dense 4-set or a 1-sparse 11-set. If it has a 1-dense 4-set, we are done. So, assume $G$ does not contain a 1-dense 4-set. Let $(A,B)$ be a bipartition of $G$. If $|A|\geq 11$ or $|B|\geq 11$, then we are done by taking $A$ or $B$, respectively, as a 1-sparse set. Hence, assume $|A|=|B|=10$. Besides, if $G$ has a vertex $w$ of degree at most one, wlog say $w\in A$, then $B\cup \{w\}$ is a 1-sparse 11-set. Thus, suppose all vertices in $G$ have degree at least two.

 Take a vertex $\omega\in G$ of maximum degree $d(w)=k$. Say wlog $\omega\in A$ and let $N(w)=\{x_1,x_2,...,x_k\}$ and $M_i=N(x_i)-\omega$. Assume wlog $|M_1|\leq|M_2|\leq\ldots\leq |M_k|$. Note that $M_i\cap M_j=\emptyset$ since $G$ has no $1$-dense $4$-set, and $|M_1|+|M_2|+\ldots+|M_k|\leq 9$. Hence, if $k\geq6$, we get $|M_1|=|M_2|=1$. Then $(A-w)\cup\{x_1,x_2\}$ is a $1$-sparse $11$-set and we are done. So, assume $k\leq 5$.

Now, if $G$ has no vertex of degree 3, then $|M_i|\in\{1,3,4\}$. Hence if $k\geq 4$, $|M_1|+|M_2|+|M_3|+|M_4|\leq 9$ implies $|M_1|=|M_2|=1$ and we are done as previously. Since $k\neq 3$, we have $k=2$ which means that  we obtain again $|M_1|=|M_2|=1$ and the result follows similarly. It follows that there exists a vertex $x$, say wlog in $A$, with $d(x)=3$. Let $N(x)=\{a,b,c\}$.

We claim $G$ has two adjacent vertices both of which has degree at most three. If one of $\{a,b,c\}$ has the degree at most three, the claim holds. Assume $d(a),d(b),d(c)\geq 4$, since $N(a)-\{x\}$, $N(b)-\{x\}$ and $N(c)-\{x\}$ are pairwise disjoint subsets of $A-\{x\}$ and $|A-\{x\}|=9$, we get $|N(a)-\{x\}|=|N(b)-\{x\}|=|N(c)-\{x\}|=3$. Take a vertex $y\in B-\{a,b,c\}$. Since $G$ has no $1$-dense $4$-set, $y$ has at most one neighbor from each one of $N(a)-\{x\}$, $N(b)-\{x\}$, $N(c)-\{x\}$, thus $d(y)\leq 3$. Therefore, there are at most $3\cdot 7 + 4\cdot 3=33$ edges between $A$ and $B$. Then, by the pigeonhole principle, there exists $z\in A-\{x\}$ with $d(z)\leq 3$. Since $|N(z)\cap\{a,b,c\}|\leq 1$, $z$ should have a neighbor in $B-\{a,b,c\}$, and so the claim holds. As a result, $G$ has two adjacent vertices $u$ and $v$ such that $d(u)\leq 3$ and $d(v)\leq 3$.

Since all degrees in $G$ are at least two, there are three cases:
	\begin{enumerate}
		\item $d(u)=d(v)=2$. Let $c$ and $a$ be the unique neighbors of $u$ and $v$ other than $\{u,v\}$, respectively. Since $d(a)\geq 2$, $a$ has at least one neighbor other than $v$, take one of them and say $d$. Similarly, take a neighbor $b\neq a$ of $c$.
		\item One of $u$ and $v$ has degree three, the other has degree two, wlog say $d(u)=3$ and $d(v)=2$. Let $c$ and $d$ be the neighbors of $u$ other than $v$. Since $d(v),d(c)\geq 2$ and $G$ has no 1-dense 4-set, $v$ and $c$ have distinct neighbors, say $a$ and $b$, respectively.
		\item $d(u)=d(v)=3$. Let $N(u)=\{v,c,d\}$ and $N(v)=\{u,a,b\}$.
	\end{enumerate}

\begin{figure}[h]
\centering
    \begin{tabularx}{1\textwidth}{*{3}{>{\centering\arraybackslash}X}}
\begin{tikzpicture}[
every edge/.style = {draw=black,very thick},
 vrtx/.style args = {#1/#2}
      circle, draw, thick, fill=white,
      minimum size=5mm, label=#1:#2}
                    ]
\node(U) [vrtx=center/$u$] at (-1, 2) {};
\node(V) [vrtx=center/$v$] at (1, 2) {};
\node(A) [vrtx=center/$a$] at (-1,1) {};
\node(C) [vrtx=center/$c$] at (1,1) {};
\node(B) [vrtx=center/$b$] at (-1,0) {};
\node(D) [vrtx=center/$d$] at (1,0) {};
\path   (U) edge (V)
(U) edge (C)
(A) edge (V)
(A) edge (D)
(B) edge (C);
\end{tikzpicture}
    \caption{Case 1}
\label{fig:case1}
    &
\begin{tikzpicture}[
every edge/.style = {draw=black,very thick},
 vrtx/.style args = {#1/#2}{%
      circle, draw, thick, fill=white,
      minimum size=5mm, label=#1:#2}
                    ]
\node(U) [vrtx=center/$u$] at (-1, 2) {};
\node(V) [vrtx=center/$v$] at (1, 2) {};
\node(A) [vrtx=center/$a$] at (-1,1) {};
\node(C) [vrtx=center/$c$] at (1,1) {};
\node(B) [vrtx=center/$b$] at (-1,0) {};
\node(D) [vrtx=center/$d$] at (1,0) {};
\path   (U) edge (V)
(U) edge (C)
(A) edge (V)
(U) edge (D)
(B) edge (C);
\end{tikzpicture}
\caption{Case 2}
\label{fig:case2}
    &
\begin{tikzpicture}[
every edge/.style = {draw=black,very thick},
 vrtx/.style args = {#1/#2}{%
      circle, draw, thick, fill=white,
      minimum size=5mm, label=#1:#2}
                    ]
\node(U) [vrtx=center/$u$] at (-1, 2) {};
\node(V) [vrtx=center/$v$] at (1, 2) {};
\node(A) [vrtx=center/$a$] at (-1,1) {};
\node(C) [vrtx=center/$c$] at (1,1) {};
\node(B) [vrtx=center/$b$] at (-1,0) {};
\node(D) [vrtx=center/$d$] at (1,0) {};
\path   (U) edge (V)
(U) edge (C)
(A) edge (V)
(U) edge (D)
(B) edge (V);
\end{tikzpicture}
\caption{Case 3}
\label{fig:case3}
    \end{tabularx}
\end{figure}
Since $G$ has no $1$-dense $4$-set, all edges present between $\{u,v,a,b,c,d\}$ in each case are given in Figures \ref{fig:case1}, \ref{fig:case2}, and \ref{fig:case3}, respectively. Now, if $G-\{u,v,a,b,c,d\}$ has a $1$-sparse $9$-set, say $S$, then $S\cup\{u,v\}$ is a 1-sparse $11$-set and we are done. So assume $G-\{u,v,a,b,c,d\}$ has no $1$-sparse $9$-set. Then we have $G-\{u,v,a,b,c,d\}\in \mathcal{T}_{14}^{\mathcal{BIP}}(1,4,9)$. Using Algorithm \ref{algo:CHECK}, we found all 73 graphs in $\mathcal{T}_{14}^{\mathcal{BIP}}(1,4,9)$, and examined all possible ways to combining these 73 graphs with one of the Cases 1, 2 and 3 to construct a bipartite graph on 20 vertices with no 1-dense 4-set.
For all the resulting graphs, we could identify a $1$-sparse $11$-set, which proves that $R_1^{\mathcal{BIP}}(4,11)=20$.
\end{proof}

Using Algorithm \ref{algo:CHECK}, we also compute some $k$-defective Ramsey numbers in bipartite graphs for $k\in\{2,3,4\}$ where these values were not known previously. In \cite{defectiveramseynumbers}, it has been stated that $R_k^{\mathcal{BIP}}(i,j)$ was open for $k+3\leq i\leq 2k+2$ and $j\geq k+2$ when $k\geq 2$.

Using Algorithm \ref{algo:CHECK}, we found some non-trivial values of $R_k^{\mathcal{BIP}}(k+3,j)$ for several $j$ values. Note that we did not use bold font to distinguish newly computed values since all values in Tables \ref{table:2-defectivebipartite}, \ref{table:3-defectivebipartite} and \ref{table:4-defectivebipartite} are new. All extremal graphs in these tables are available in our github account \cite{github}. Unlike for perfect graphs, we check bipartiteness just before checking forbidden defective sets since bipartite graphs can be recognized very efficiently as compared to perfect graphs. We note that in these calculations, we obtained defective Ramsey numbers up to 15 in less than five minutes, so the following tables can be extended by allowing the algorithm to run longer.

	\begin{theorem}
		The following hold where we denote the number of extremal graphs in parenthesis:

\begin{table}[H]\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|}
					\hline
					\textbf{$R_2^{\mathcal{BIP}}(i,j)$} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8}  \\
					\hline

					\textbf{5} & 5(1) & 6(4) & 8(1) & 10(2)& 11(56)\\
					\hline

					\textbf{6} & 5(1) & 7(3) & 9(1) & 11(35) & 14(3)\\
					\hline

					\textbf{7} & 5(1) & 9(2) & 11(6) & 13(249) & \\
					\hline

					\textbf{8} & 5(1) & 9(2) & 11(6) & 13(249) & \\
					\hline

				\end{tabular}
			}
			\caption{2-Defective Ramsey Numbers in Bipartite Graphs}
			\label{table:2-defectivebipartite}
		\end{table}
		\begin{table}[H]

			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_3^{\mathcal{BIP}}(i,j)$} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9}& \textbf{10}\\
					\hline

					\textbf{6} & 6(1) & 7(5) & 9(1) & 10(8) & 12(1) & 13(9) \\
					\hline

					\textbf{7} & 6(1) & 8(2) & 10(1) & 12(2)&14(26)&\\
					\hline

					\textbf{8} & 6(1) & 8(2) & 10(10) & 13(2)& 15(423)&\\
					\hline

					\textbf{9} & 6(1) & 8(2) & 13(5) & 15(40)& &\\
					\hline
				\end{tabular}

			}

			\caption{3-Defective Ramsey Numbers in Bipartite Graphs}
			\label{table:3-defectivebipartite}

		\end{table}
		\begin{table}[H]
			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_4^{\mathcal{BIP}}(i,j)$} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10}  \\
					\hline

					\textbf{7} & 7(1) & 8(6) & 10(1) & 11(7) & 12(34) \\
					\hline

					\textbf{8} & 7(1) & 9(2) & 11(1) & 12(29) & 14(16) \\
					\hline

					\textbf{9} & 7(1) & 9(2) & 11(7) & 13(19) & 15(133) \\
					\hline

					\textbf{10} & 7(1) & 9(2) & 11(7) & 13(70) &  \\
					\hline

				\end{tabular}
			}

			\caption{4-Defective Ramsey Numbers in Bipartite Graphs}
			\label{table:4-defectivebipartite}
		\end{table}

\end{theorem}

As supported by the above tables, we obtain the following result which settles one of the aforementioned open cases for bipartite graphs pointed out in \cite{defectiveramseynumbers}.
\begin{theorem}
For all $k\geq 1$ and $k+3\leq i\leq 2k+2$, we have $R_k^{\mathcal{BIP}}(i,k+2)=k+3$.
\end{theorem}
\begin{proof}
Consider a complete bipartite graph with one vertex in one part and $k+1$ vertices in the other part. It has no $k$-dense $i$-set (since there are less than $i$ vertices) and no $k$-sparse $(k+2)$-set (since there is a vertex of degree $k+1$). Now, take a bipartite graph of order at least $k+3$. If one part contains $k+2$ vertices, they form a $k$-sparse $(k+2)$-set. So assume each part contains at most $k+1$ vertices. Then, any subset of $k+2$ vertices containing at most $k$ vertices from each side forms a $k$-sparse set.
\end{proof}

	\section{Defective Ramsey Numbers in Chordal Graphs}	\label{sec:chordal}

	\noindent In this section, we use Algorithm \ref{algo:CHECK} as described in Remark \ref{rem:algo} to compute new defective Ramsey numbers in chordal graphs and determine their extremal graphs. A graph is \textit{chordal} if it contains no induced  cycle of length four or more. Some characteristics of chordal graphs enables us to generate the set $\mathcal{T}_{n+1}^\mathcal{CH}(k,i,j)$ in an efficient way (from the input set $\mathcal{T}_n^\mathcal{CH}(k,i,j)$). A vertex is called \textit{simplicial} if its neighborhood induce a compete graph. A \textit{perfect elimination ordering} $\sigma$ in a graph is an ordering of the vertices of the graph such that, every vertex $v$ is simplicial in the graph induced by the vertices coming after $v$ in $\sigma$. It is known that every chordal graph has a simplicial vertex; moreover, a graph is chordal if and only if it has a perfect elimination ordering \cite{chordal_peo}.

\begin{lemma}\label{lem:chordal}
The set $\mathcal{T}_{n+1}^\mathcal{CH}(k,i,j)$ can be generated by adding only simplicial vertices to the graphs in the set $\mathcal{T}_n^\mathcal{CH}(k,i,j)$ in line \ref{line:allS} and without checking chordality in line \ref{line:G} of Algorithm \ref{algo:CHECK}.
\end{lemma}
\begin{proof}
Let $G \in \mathcal{T}_n^\mathcal{CH}(k,i,j)$ and $G'$ be a graph in $\mathcal{T}_{n+1}^\mathcal{CH}(k,i,j)$ such that $G'$ contains $G$ as an induced subgraph. Let $x$ be the (unique) vertex in $V(G')\setminus V(G)$. Then, for any vertex $y\neq x$ in $G'$, the graph $G' -y \in \mathcal{T}_n^\mathcal{CH}(k,i,j)$ (since otherwise $G'$ would contain a $k$-dense $i$-set or $k$-sparse $j$-set, thus not belong to $\mathcal{T}_{n+1}^\mathcal{CH}(k,i,j)$). Since every chordal graph has a simplicial vertex, this also holds for a simplicial vertex $y\neq x$ in $G'$ (whose existence is guaranteed since $G$ is chordal), implying that all graphs in $\mathcal{T}_{n+1}^\mathcal{CH}(k,i,j)$ can be obtained by adding a simplicial vertex to a graph in $\mathcal{T}_{n}^\mathcal{CH}(k,i,j)$. Accordingly, in line \ref{line:allS} of Algorithm \ref{algo:CHECK}, it is sufficient to select only subsets of vertices forming a clique as the neighborhood of the newly added vertex. Moreover, all graphs constructed in this way are chordal since a perfect elimination ordering $\sigma$ of $G$, preceded by the newly added simplicial vertex $y$ is a perfect elimination order for $G'$. Therefore, the chordality check in line \ref{line:G} would always return a positive answer, thus it can be omitted.
\end{proof}

Following Lemma \ref{lem:chordal}	, we can assume that the newly added vertex in line \ref{line:allS} is simplicial. Accordingly, we only consider subsets of vertices forming a clique when generating new graphs. Moreover, we do not check chordality in line \ref{line:G} as it is already ensured by Lemma \ref{lem:chordal}. We report the values of $k$-defective Ramsey numbers in chordal graphs for $k\in \{1,2,3,4\}$ in Tables \ref{table:1-defectivechordal}, \ref{table:2-defectivechordal},  \ref{table:3-defectivechordal} and  \ref{table:4-defectivechordal} respectively. We run Algorithm \ref{algo:CHECK} at most 12 hours to obtain the values in Table  \ref{table:1-defectivechordal} whereas each one of the   values (up to 15) in the remaining tables are obtained in at most 5 hours.

	\begin{theorem}
		The following hold where we denote the number of extremal graphs in parenthesis:
\end{theorem}

	\begin{table}[H]\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_1^{\mathcal{CH}}(i,j)$} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10} & \textbf{11} \\
					\hline

					\textbf{3} & 3(2) & 4(2) & 5(3) & 6(3) & 7(4) & 8(4) & 9(5) & 10(5) &11(6)\\
					\hline

					\textbf{4} & 4(2) & 6(1) & 8(1) & 10(1) & 12(1) & 14(1) & 16(1) & 18(1) & 20(1) \\
					\hline

					\textbf{5} & 5(2) & 7(4) & 10(4) & 12(44) & 15(18) &  & &&\\
					\hline

					\textbf{6} & 6(2) & 8(8) & 12(17) & 14(1397) & & &&&\\
					\hline

					\textbf{7} & 7(2) & 10(1) & 14 (68) & & & &&&\\
					\hline

					\textbf{8} & 8(2) & 11(2) & 16(293) & & & &&&\\
					\hline

					\textbf{9} & 9(2) & 12(4) & 18(1245) & & & & &&\\
					\hline

				\end{tabular}
			}

			\caption{1-Defective Ramsey Numbers in Chordal Graphs}
			\label{table:1-defectivechordal}

	\end{table}
	\begin{table}[H]

			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_2^{\mathcal{CH}}(i,j)$} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10} \\
					\hline

					\textbf{4} & 4(4) & 5(4) & 6(4) & 7(4) & 8(4) & 9(4) &10(4) \\
					\hline

					\textbf{5} & 5(4) & 7(2) & 8(11) & 9(101) & 11(66) & 13(24)&\\
					\hline

					\textbf{6} & 6(4) & 8(8) & 10(2) & 12(45) & 14(92) & &\\
					\hline

					\textbf{7} & 7(4) & 9(22) & 11(50) & 14(316) & & &\\
					\hline

					\textbf{8} & 8(4) & 11(1) & 12(469) & & & &\\
					\hline

					\textbf{9} & 9(4) & 12(4) & 14(13) & & & & \\
					\hline

					\textbf{10} & 10(4) & 13(11) & 15(194) &  & & &\\
					\hline

				\end{tabular}
			}

			\caption{2-Defective Ramsey Numbers in Chordal Graphs}
			\label{table:2-defectivechordal}

	\end{table}
	\begin{table}[H]

			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_3^{\mathcal{CH}}(i,j)$} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10} \\
					\hline

					\textbf{5} & 5(10) & 6(10) & 7(11) & 8(11) & 9(12) & 10(12)\\
					\hline

					\textbf{6} & 6(10) & 8(4) & 9(24) & 10(123) & 12(2) & 13(43) \\
					\hline

					\textbf{7} & 7(10) & 9(19) & 11(4) & 12(151) & 14(2) & \\
					\hline

					\textbf{8} & 8(10) & 10(62) & 12(124) & 14(7) & &\\
					\hline

					\textbf{9} & 9(10) & 12(2) & 13(1846) & & & \\
					\hline

				\end{tabular}
			}

			\caption{3-Defective Ramsey Numbers in Chordal Graphs}
			\label{table:3-defectivechordal}

	\end{table}
	\begin{table}[H]
			\centering
			\scalebox{1}{
				\begin{tabular}{|C|C|C|C|C|C|C|C|C|C|C|}
					\hline
					\textbf{$R_4^{\mathcal{CH}}(i,j)$} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10} & \textbf{11} \\
					\hline

					\textbf{6} & 6(27) & 7(27) & 8(27) & 9(27) & 10(27) & 11(27)\\
					\hline

					\textbf{7} & 7(27) & 9(10) & 10(64) & 11(360) & 13(4) & 14(24)\\
					\hline

					\textbf{8} & 8(27) & 10(53) & 12(8) & 13(364) & &\\
					\hline

					\textbf{9} & 9(27) & 11(207) & 13(322) & & &\\
					\hline

					\textbf{10} & 10(27) & 13(4) &  &  & &\\
					\hline

				\end{tabular}
			}

			\caption{4-Defective Ramsey Numbers in Chordal Graphs}
			\label{table:4-defectivechordal}
		\end{table}

In what follows, we prove that the pattern that we observe for $R_k^{\mathcal{CH}}(k+2,j)$, $R_k^{\mathcal{CH}}(i,k+2)$ and  $R_1^{\mathcal{CH}}(4,j)$ in Tables \ref{table:1-defectivechordal}, \ref{table:2-defectivechordal}, \ref{table:3-defectivechordal}, \ref{table:4-defectivechordal} holds for all $i$ and $j$. Moreover, we describe the unique extremal graph for $R_1^{\mathcal{CH}}(4,j)$ for $4\leq j \leq 11$.

\begin{lemma}\label{lemmaempty} \cite{defectiveramseynumbers}
Let $\mathcal{G}$ be a graph class containing all empty graphs. Then,
$$R_{k}^{\mathcal{G}}(k+2, j)=j \text { for all } j \geq k+2.$$
\end{lemma}

Since an empty graph is chordal, Lemma \ref{lemmaempty} implies the following:
\begin{remark}
 $R_k^{\mathcal{CH}}(k+2,j)=j$ for all $k\geq 1$ and $j\geq k+2$.
\end{remark}

By taking the complementary class, the following is implied by Lemma \ref{lemmaempty}.

\begin{corollary}\label{lemmacomplete}
Let $\mathcal{G}$ be a graph class containing all complete graphs. Then,
$$R_{k}^{\mathcal{G}}(i, k+2)=i \text { for all } i \geq k+2.$$
\end{corollary}

Since a complete graph is chordal, Lemma \ref{lemmacomplete} implies the following:
\begin{remark}
 $R_k^{\mathcal{CH}}(i,k+2)=i$ for all $k\geq 1$ and $i\geq k+2$.
\end{remark}

\begin{theorem}
$R_1^{\mathcal{CH}}(4,j)=2j-2$ for all $j\geq 3$.
\end{theorem}

\begin{proof}
Let $G$ be a chordal graph on at least $2j-2$ vertices. If $G$ has a 1-dense 4-set, we are done. If not, $G$ does not contain a cycle of size 4 (as a partial subgraph) since it is 1-dense 4-set. Then, all 2-connected components of $G$ are either triangles or edges; thus, it is a \textit{cactus} graph as any two cycles of it have no edge in common. Let $\mathcal{CA}$ denote the class of cactus graphs. From  \cite{defectiveramseynumbers}, we know that $R_1^{\mathcal{CA}}(4,j)=2j-2$ for all $j\geq3$. It follows that $G$ contains a 1-sparse $j$-set, and we are done for all $j\geq4$. We conclude the proof with the extremal graph (see Figure \ref{fig:extremalforChordal}) that has been constructed for cactus graphs in \cite{defectiveramseynumbers}. This is a graph on $2j-3$ vertices with neither 1-dense 4-set nor 1-sparse $j$-set for all $j\geq3$.
\end{proof}

\def4pt{4pt}
\def1cm{1cm}
\tikzset{c/.style={draw,circle,fill=white,minimum size=4pt,inner sep=0pt,
		anchor=center},
	d/.style={draw,circle,fill=black,minimum size=4pt,inner sep=0pt, anchor=center}}

\begin{figure}[h]
	\begin{center}

		\begin{tikzpicture}

		\node[d] at (0,2) {};
		\node[d] at (1,1) {};
		\node[d] at (-1,1) {};
		\draw (0,2) to (1,1)
		(0,2) to (-1,1)
		(-1,1) to (1,1);

		\node[d] at (2,2) {};
		\node[d] at (3,1) {};
		\draw (1,1) to (2,2)
		(1,1) to (3,1)
		(3,1) to (2,2);

		\node[thick,minimum size=3cm] at (4,1) {.....};

		\node[d] at (6,2) {};
		\node[d] at (5,1) {};
		\node[d] at (7,1) {};
		\draw (5,1) to (6,2)
		(7,1) to (6,2)
		(7,1) to (5,1);

		\node[d] at (8,1) {};
		\draw (7,1) to (8,1);

		\node[d] at (-2,1) {};
		\draw (-1,1) to (-2,1);

		\draw [decorate,decoration={brace,amplitude=10pt},xshift=0pt,yshift=4pt]
		(-0.1,2) -- (6.1,2) node [black,midway,yshift=0.6cm]
		{\footnotesize $j-3$};

		\end{tikzpicture}
		\vspace{-1.5cm}
	\end{center}
	\caption{An extremal graph for $R_1^{\mathcal{CH}}(4,j)=2j-2$ for $j\geq 3$ with $2j-3$ vertices.}\label{fig:extremalforChordal}
\end{figure}

According to Table \ref{table:1-defectivechordal} obtained by Algorithm \ref{algo:CHECK}, the extremal graph described in Figure \ref{fig:extremalforChordal} is the unique extremal graph for  $R_1^{\mathcal{CH}}(4,j)=2j-2$ for all $4\leq j\leq 11$. However, we do not know if this trend continues for all $j \geq 12$.

	\section{Defective Cocolorings} \label{sec:coco}

	\noindent  In this section, we study the parameter $c_k^{\mathcal{G}}(m)$ for perfect graphs and cographs. Recall that  $c_k^{\mathcal{G}}(m)$ is the maximum integer $n$ such that for every graph $G$ on $n$ vertices in the graph class $\mathcal{G}$, the vertices of $G$ can be partitioned into $m$ subsets where each subset is $k$-defective (either $k$-dense or $k$-sparse). In other words, every graph in a class $\mathcal G$ on at most $c_k^{\mathcal{G}}(m)$ vertices can be $k$-defectively cocolored with at most $m$ colors. In this section, an \textit{extremal graph for $c_k^{\mathcal{G}}(m)=n$} is a graph of order $n+1$ that belongs to the class $\mathcal G$ and whose vertices can not be partitioned into at most $m$ sets each of which is $k$-defective.

We start with some general observations that will be useful when studying defective cocolorings in perfect graphs and in cographs.  Since general graphs contain all graph classes, the following is an immediate consequence :
\begin{remark}
For any graph class $\mathcal G$ and all integers $k\geq 0$, $m\geq 1$, we have $c_k(m)\leq c^{\mathcal{G}}_k(m)$.
\end{remark}

This remark together with previously known results guide us through our research. From \cite{defectiveparameterTinazAhu}, we know $c_k(1)=k+2$, $c_1(2)=7$, $c_2(2)=10$, and $c_1(3)=12$. In the same paper, it is also conjectured that $c_k(2)=3k+4$.
We first generalize the formula $c_k(1)=k+2$ to graph classes.

\begin{lemma}\label{lem:c_k_1_G}
For all integers $k\geq 0$ and all graph classes $\mathcal G$ containing $K_{1,k+2}$, we have $c_k^{\mathcal{G}}(1)=k+2$.
\end{lemma}
\begin{proof}
The result follows by observing that $K_{1,k+2}\in\mathcal{G}$ is not $k$-defective and each set of size $k+2$ is $k$-defective.
\end{proof}

In \cite{defectiveparameterTinazAhu}, Straight's formula from \cite{StraightsFormula} has been generalized to the defective version as follows: $c_k(m-1)+t\leq c_k(m)\leq c_k(m-1)+(m+1)(k+1)$ where $t$ is any positive integer satisfying $c_k(m-1)+t\geq R_k(t,t)$. Next, we adapt the lower bound to graph classes.

\begin{lemma}\label{lem:Straightlowerbound}
If $c_k^{\mathcal{G}}(m-1)+t\geq R_k^{\mathcal{G}}(t,t)$ for some integers $k\geq 0$, $m\geq 2$, $t\geq 1$ and graph class $\mathcal{G}$, then we have $c_k^{\mathcal{G}}(m-1)+t\leq c_k^{\mathcal{G}}(m)$.
\end{lemma}

\begin{proof}
Let us take a graph $G\in\mathcal{G}$ on $c_k^{\mathcal{G}}(m-1)+t$ vertices where $c_k^{\mathcal{G}}(m-1)+t\geq R_k^{\mathcal{G}}(t,t)$. By definition of $R_k^{\mathcal{G}}(t,t)$, $G$ has a $k$-defective $t$-set, say $T$. Now, $G-T$ has $c_k^{\mathcal{G}}(m-1)$ vertices, so it can be colored with $m-1$ colors where each color class is a $k$-defective set, which completes the proof.
\end{proof}

Secondly, we show that the upper bound of the above inequality can be adapted to graph classes with an additional property on the class. We use the same idea as in \cite{defectiveparameterTinazAhu} with a slight modification, which yields an improvement.

\begin{lemma}\label{lem:Straightupperbound}
	Let $\mathcal{G}$ be a graph class that is closed under taking the disjoint union with any clique. Then we have $c_k^{\mathcal{G}}(m)\leq c_k^{\mathcal{G}}(m-1)+m(k+1)+1$.
\end{lemma}

\begin{proof}
	Take a graph class $\mathcal{G}$ with desired property. Let us construct a graph lying in $\mathcal{G}$ and having $c_k^{\mathcal{G}}(m-1)+m(k+1)+2$ vertices which cannot be $k$-defectively cocolored  using at most $m$ colors. By definition of $c_k^{\mathcal{G}}(m-1)$, there is a graph $H\in \mathcal G$ on $1+c_k^{\mathcal{G}}(m-1)$ vertices that cannot be partitioned into $m-1$ many $k$-defective sets. Consider the disjoint union of $K_{1+m(k+1)}$ and $H$, say $G$. Note that $G\in\mathcal{G}$ by definition of $G$, and it has $c_k^{\mathcal{G}}(m-1)+m(k+1)+2$ vertices. Assume $G$ can be partitioned into $m$ subsets such that each subset is $k$-defective. By the pigeonhole principle, there is a subset $S$ for which $|S\cap K_{1+m(k+1)}|\geq k+2$. Since $S$ is a $k$-defective set and each vertex in $S\cap K_{1+m(k+1)}$ has at least $k+1$ neighbors in $S$, it follows that $S$ is not $k$-sparse. So, $S$ is $k$-dense; therefore it contains no vertex from $H$ or else it would miss at least $k+2$ vertices of $S \cap K_{1+m(k+1)}$. As a result, $G-S$ contains $H$ and can to be partitioned into $m-1$ many $k$-defective sets, which is a contradiction since $H$ itself already requires at least $m$ colors.
\end{proof}

Now, we are ready to discuss the defective cocolorings in graph classes.

\subsection{Perfect Graphs}\label{sec:cocoperfect}

Firstly, we establish the formula for the classical cocoloring in perfect graphs where the defectiveness level is zero.
	\begin{theorem}\label{thm:c_0_m_PG}
		For all integers $m\geq 1$, we have $c_0^{\mathcal{PG}}(m)=\dfrac{m(m+3)}{2}$.
	\end{theorem}

	\begin{proof}
		Firstly, consider the disjoint union of $K_i$ where $i\in\{1,2,...,m+1\}$. It is a perfect graph on $1+2+...+(m+1)=\dfrac{(m+1)(m+2)}{2}=\dfrac{m(m+3)}{2}+1$ vertices. It can be easily seen that this graph cannot be partitioned into $m$ cliques or an independent sets. Secondly, let us show that every perfect graph on $\dfrac{m(m+3)}{2}$ vertices can be colored with $m$ colors such that each color class is either a clique or an independent set. We prove this by induction on $m$. The claim is trivial for $m=1$. Assume it holds for smaller values of $m$, and take a perfect graph $G$ on $\dfrac{m(m+3)}{2}$ vertices. If $\chi(G)\leq m$, then we are done. Assume $\chi(G)\geq m+1$, we get $\omega(G)\geq m+1$ since $G$ is perfect. Let us color a maximum clique with color $m$. Since the number of remaining vertices is at most $\dfrac{m(m+3)}{2}-(m+1)=\dfrac{(m-1)(m+2)}{2}$, they can be cocolored with $m-1$ colors by the induction hypothesis, hence we are done.
	\end{proof}

We can also note that we have $c_k^{\mathcal{PG}}(1)=k+2$ by Lemma \ref{lem:c_k_1_G} since all star graphs are perfect. We proceed with the computation of our parameter for non-trivial cases. We use a combination of theoretical analysis and computer assisted approach.

	\begin{theorem}\label{thm:c_1_2_PG}
		We have $c_1^{\mathcal{PG}}(2)=7$ with 24 extremal graphs.
	\end{theorem}

	\begin{proof}
		We note that $c_1^{\mathcal{PG}}(2)\geq c_1(2)=7$. Now, let us show that there are $24$ extremal graphs; that is perfect graphs on 8 vertices that can not be 1-defectively cocolored using at most 2 colors. Take a perfect graph on 8 vertices. If it has a 1-defective set of size 5, then two colors are enough since any set of three vertices is 1-defective. So any extremal graph on 8 vertices is free of 1-defective 5-sets, in other words, it belongs to the set  $\mathcal{T}_{8}^\mathcal{PG}(1,5,5)$. Accordingly, we generated the set $\mathcal{T}_{8}^\mathcal{PG}(1,5,5)$ using Algorithm \ref{algo:CHECK} starting with the single vertex graph as input. For each one of the resulting 824 graphs in $\mathcal{T}_{8}^\mathcal{PG}(1,5,5)$, we checked all possible partitions into two sets of four vertices each, and found that exactly 24 many of them cannot be partitioned into two 1-defective sets. The list of these 24 extremal graphs for $c_1^{\mathcal{PG}}(2)=7$ can be found in our github account \cite{github}.
	\end{proof}

Even though we could not find the number of extremal graphs, we get the exact value for $k=m=2$.

	\begin{theorem}\label{thm:c_2_2_PG}
		We have $c_2^{\mathcal{PG}}(2)=11$.
	\end{theorem}

	\begin{proof}
	By Lemmas \ref{lem:c_k_1_G} and \ref{lem:Straightupperbound}, we have $c_2^{\mathcal{PG}}(2)\leq c_2^{\mathcal{PG}}(1)+7=11$. Let us now show that all perfect graphs on 11 vertices can be partitioned into two $2$-defective sets. Take a perfect graph $G$ on $11$ vertices. Since $R_2^{\mathcal{PG}}(6,6)=10$ from \cite{defectiveramseynumbers}, $G$ has a $2$-defective $6$-set, say $S$.  If $G-S$ is $2$-defective, we are done. So, assume $G-S$ is not 2-defective. Let $\mathcal{A}$ be the set of all perfect graphs on 6 vertices that are $2$-defective, and $\mathcal{B}$ be the set of all graphs on 5 vertices that are not $2$-defective.  Then the vertex set of $G$ can be partitioned into two parts, one belonging to $\mathcal{A}$ and the other one belonging to $\mathcal{B}$. Using a computer, we enumerated all possible graphs obtained as a combination of two graphs $A$ and $B$ for some $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We checked all such graphs and concluded that they can all be partitioned into two 2-defective sets, hence the desired result.

\end{proof}

Next, we investigate $c_1^{\mathcal{PG}}(3)$ and $c_3^{\mathcal{PG}}(2)$. For these parameters, we provide bounds with two or three possible values, leaving the computation of the exact value open.

\begin{theorem}\label{thm:c_bounds_PG}
We have $13\leq c_1^{\mathcal{PG}}(3)\leq 14$ and $12\leq c_3^{\mathcal{PG}}(2)\leq 14$.
\end{theorem}

\begin{proof}
 Using Lemma \ref{lem:Straightupperbound} and Theorem \ref{thm:c_1_2_PG}, we get $c_1^{\mathcal{PG}}(3)\leq c_1^{\mathcal{PG}}(2)+7=14$. For the lower bound, we note that $c_1(3)=12$ from \cite{defectiveparameterTinazAhu}, which implies $12\leq c_1^{\mathcal{PG}}(3)$. Then, we show by computer enumeration that  $c_1^{\mathcal{PG}}(3)\neq12$. Indeed, $c_1^{\mathcal{PG}}(3)=12$ would imply that there is a perfect graph $G$ on 13 vertices which cannot be partitioned into three $1$-defective sets. Besides, since $R_1^{\mathcal{PG}}(5,5)=13$ from \cite{1defectiveperfectTinazOylumJohn}, $G$ has a $1$-defective $5$-set , say $J$. Then $G-J$ is a perfect graph on 8 vertices which can not be 1-defectivelty cocolored with 2 colors. It follows from Theorem \ref{thm:c_1_2_PG} that $G-J$ is one of the 24 extremal graphs for $c_1^{\mathcal{PG}}(2)=7$. Hence, $G$ is obtained as a combination of a 1-defective 5-set and one of these 24 extremal graphs. We checked all possible combinations (which took more than a month) and concluded that all such graphs can be partitioned into three $1$-defective sets, which yields $c_1^{\mathcal{PG}}(3)\neq12$.

As for $c_3^{\mathcal{PG}}(2)$, we have $R_3^{\mathcal{PG}}(7,7)=11$ from \cite{defectiveramseynumbers} and $c_3^{\mathcal{PG}}(1)=5$ from Lemma \ref{lem:c_k_1_G}. Thus, the inequality $c_k^{\mathcal{PG}}(m-1)+t\geq R_k^{\mathcal{PG}}(t,t)$ holds for $k=3$, $m=2$ and $t=7$, which gives $12=c_3^{\mathcal{PG}}(1)+7\leq c_3^{\mathcal{PG}}(2)$ by Lemma \ref{lem:Straightlowerbound}. On the other hand, by using Lemmas \ref{lem:Straightupperbound} and \ref{lem:c_k_1_G}, we get $c_k^{\mathcal{PG}}(2)\leq c_k^{\mathcal{PG}}(1)+1+2(k+1)=3k+5$ for all $k$, which implies $c_3^{\mathcal{PG}}(2)\leq 14$.
\end{proof}

\subsection{Cographs} \label{sec:cococograph}
\textit{Cographs}, denoted by $\mathcal{CO}$, is the class of graphs containing no induced path on four vertices. We start by noting that for the classical case ($k=0$), we have $c_0^{\mathcal{CO}}(m)=c_0^{\mathcal{PG}}(m)$ since the extremal graph constructed in the proof of Theorem \ref{thm:c_0_m_PG} is also a cograph and since $\mathcal{CO}\subset \mathcal{PG}$ we have $c_k^{\mathcal{PG}}(m)\leq c_k^{\mathcal{CO}}(m)$ for all $k$ and $m$.

\begin{remark}\label{rem:c_0_CO}
We have $c_0^{\mathcal{CO}}(m)=\dfrac{m(m+3)}{2}$.
\end{remark}

\noindent Secondly, we have $c_k^{\mathcal{CO}}(1)=k+2$ from Lemma \ref{lem:c_k_1_G}. Then, the next candidates for examination are $c_1^{\mathcal{CO}}(2)$ and $c_2^{\mathcal{CO}}(2)$. Both values can be found as a natural consequence of results in perfect graphs.

\begin{theorem}\label{thm:c_results_cographs}
We have $c_1^{\mathcal{CO}}(2)=8$ and $c_2^{\mathcal{CO}}(2)=11$.
\end{theorem}

\begin{proof}
Note that $c_1^{\mathcal{CO}}(2)\leq c_1^{\mathcal{CO}}(1)+5=8$ from Lemma \ref{lem:Straightupperbound}. Moreover, none of the extremal graphs in Theorem \ref{thm:c_1_2_PG} is a cograph, which implies $c_1^{\mathcal{CO}}(2)\geq 8$. On the other hand, since $c_2^{\mathcal{PG}}(2)=11$ from Theorem \ref{thm:c_2_2_PG}, we get $11\leq c_2^{\mathcal{CO}}(2)\leq c_2^{\mathcal{CO}}(1)+7=11$ by using Lemma \ref{lem:Straightupperbound}, so the result follows.
\end{proof}

\noindent Now, we have $c_0^{\mathcal{CO}}(2)=5$, $c_1^{\mathcal{CO}}(2)=8$ and $c_2^{\mathcal{CO}}(2)=11$. In what follows, we show that this trend holds for all $k\geq 3$.

\begin{theorem}
We have $c_k^{\mathcal{CO}}(2) = 3k+5$ for $k\geq 3$.
\end{theorem}

\begin{proof}
We first observe that we have $c_k^{\mathcal{CO}}(2)\leq c_k^{\mathcal{CO}}(1)+2(k+1)+1$ from Lemma \ref{lem:Straightupperbound}. Since $c_k^{\mathcal{CO}}(1)=k+2$, we obtain $c_k^{\mathcal{CO}}(2) \leq 3k+5$ for $k\geq 3$.

Let $G$ be a cograph of order $3k+5$. Since the complement of a cograph $G$ is also a cograph, and exactly one of $G$ or its complement $\overline{G}$ is disconnected \cite{cograph}, we assume without loss of generality that $G$ is disconnected. If $G$ has a $k$-defective $(2k+3)$-set, we are done since every $(k+2)$-set is $k$-defective. Therefore, assume $\alpha_k(G)\leq 2k+2$ and $\omega_k(G)\leq 2k+2$, and let $t$ be the number of connected components of $G$ whose size is at least $k+1$. If $t\geq 3$, then choosing $k+1$ vertices from each of these components forms a $k$-sparse $(3k+3)$-set, which is a contradiction. Besides, if $t=0$, then all vertices of $G$ is a $k$-sparse set. Thus, we need to examine two cases: $t=1$ and $t=2$.

Suppose $t=2$, and let $U$ and $V$ be the components of $G$ with $|U|\geq k+1$ and $|V|\geq k+1$. Now, if $G-(U\cup V)\neq \emptyset$, take a vertex $x\in G-(U\cup V)$ and choose $k+1$ vertices from each of $U$ and $V$. This forms a $k$-sparse $(2k+3)$-set, which is a contradiction. Hence, we can assume $G$ has only two components $U$ and $V$. Now, if one of $U$ and $V$ has a $k$-sparse $(k+2)$-set, without loss of generality say $U$, then a $k$-sparse $(k+2)$-set of $U$ together with $k+1$ vertices from $V$ form a $k$-sparse $(2k+3)$-set, a contradiction. Since  $R_k^{\mathcal{CO}}(i,k+2)=i$ for all $i\geq k+2$ (by \cite{defectiveramseynumbers}), both $U$ and $V$ are $k$-dense, and we are done.

Suppose $t=1$, and let $H$ be the connected component of $G$ with $|H|\geq k+1$. Since $G$ is disconnected, $G-H$ is non-empty, and it forms a $k$-sparse set. Moreover, if we can take a $k$-sparse set from $H$ and add it into $G-H$, we get a $k$-sparse set, too. Hence, we have $\alpha_k(H)+|G-H|\leq 2k+2$, which implies $\alpha_k(H)\leq |H|-k-3$. Similarly, consider the largest $k$-dense subset of $H$. If its size is at least $|H|-k-1$, we can add the remaining at most $k+1$ vertices of $H$ into $G-H$ to obtain a $k$-sparse set. This gives a $k$-defective 2-cocoloring of $G$, and we are done. So, assume $\omega_k(H)\leq |H|-k-2$. It follows that $\alpha_k(H)+\omega_k(H)\leq 2|H|-2k-5$, and since $|H|\leq 3k+4$, we have $\alpha_k(H)\leq 2k+1$ and $\omega_k(H)\leq 2k+2$. Since $k+2\leq \alpha_k(H)+1 \leq 2k+2$, if $\omega_k(H)\neq 2k+2$ then we have $k+2 \leq \omega_k(H)+1 \leq 2k+2$ and thus $R_k^{\mathcal{CO}}(\alpha_k(H)+1,\omega_k(H)+1)=\alpha_k(H)+\omega_k(H)-k$ from \cite{defectiveramseynumbers}. However, this gives $$1+|H|\leq R_k^{\mathcal{CO}}(\alpha_k(H)+1,\omega_k(H)+1)\leq 2|H|-3k-5,$$ implying that $|H| \geq 3k+6$, a contradiction with $|H|\leq 3k+4$. As a result, we have $\omega_k(H)=2k+2$, implying that $|H|-k-2\geq 2k+2$ and thus $|H|=3k+4$.

So, $H$ is a connected cograph on $3k+4$ vertices with $\omega_k(H)=2k+2$, and $G-H$ is an isolated vertex. Let us consider $\overline{G}$ which consists of the join of the disconnected cograph $F=\overline{H}$ having $\alpha_k(F)=2k+2$ with a single vertex, say $u$. We will show that $\overline{G}$ can be partitioned into two $k$-defective sets.

Let $s$ be the number of connected components of $F$ whose size is at least $k+1$. Since $\alpha_k(F)=2k+2$, we clearly get $s\in\{1,2\}$. If $s=2$, let $U_1$ and $V_1$ be components of $F$ with $|U_1|, |V_1|\geq k+1$. Since any subset of $k+1$ vertices from each of $U_1$ and $V_1$ form a $k$-sparse set and $\alpha_k(F)=2k+2$, we have $\alpha_k(U_1)=\alpha_k(V_1)=k+1$ and $F-(U_1\cup V_1)=\emptyset$. Again, by using $R_k^{\mathcal{CO}}(i,k+2)=i$ for all $i\geq k+2$ (by \cite{defectiveramseynumbers}), it follows that both $U_1$ and $V_1$ are $k$-dense. Since $u$ is adjacent to all vertices in $\overline{G}$, $U_1\cup \{u\}$ is $k$-dense too, thus $\overline{G}$ can be partitioned into two $k$-defective sets.

Suppose $s=1$ and let $A$ be the connected component of $F$ with $|A|\geq k+1$. Let $S$ be a largest $k$-sparse set in $A$, $K$ be a largest $k$-dense set in $A$, and write $B=S-A$. Notice that if $|A|-|S|\leq k+1$, then $(A-S)\cup \{u\}$ is $k$-dense and $S\cup B$ is $k$-sparse in $\overline{G}$. Similarly, if $|A|-|K|\leq k+1$, then $(A-K)\cup B$ is $k$-sparse and $K\cup \{u\}$ is $k$-dense. In both cases  $\overline{G}$ can be partitioned into two $k$-defective sets. Therefore, assume $|A|-\alpha_k(A),|A|-\omega_k(A)\geq k+2$, which gives $2|A|\geq \alpha_k(A)+\omega_k(A)+2k+4$. Moreover, since $B$ is nonempty, we get $|A|\leq 3k+3$ and so $k+1\leq \alpha_k(A),\omega_k(A)\leq 2k+1$. Hence, since $R_k^{\mathcal{CO}}(\alpha_k(A)+1,\omega_k(A)+1)=\alpha_k(A)+\omega_k(A)-k$ from \cite{defectiveramseynumbers}, we get $1+|A|\leq \alpha_k(A)+\omega_k(A)-k\leq 2|A|-3k-4$, which leads $|A|\geq 3k+5$ and so a contradiction. As a result, $\overline{G}$ can be partitioned into two $k$-defective sets, so can be $G$, we are done.
\end{proof}

\section{Conclusion}
In this work, we investigated the computation of defective Ramsey numbers and the parameter  $c_k^{\mathcal{G}}(m)$ in restricted graph classes. We obtained several results for perfect graphs, bipartite graphs, chordal graphs and cographs. Our approach combines efficient graph generation methods with classical direct proof techniques. We believe that the generic framework that we offer for efficiently generating structured graphs with desired properties is a promising approach in Ramsey theory, and more broadly in extremal graph theory.

\section*{Acknowledgments}
The authors acknowledge the support of the The Scientific and Technological Research Council of Turkey (T{\"U}B\.{I}TAK) Grant no:118F397.

\section*{Declarations}

\paragraph{Funding}
This work has been supported by The Scientific and Technological Research Council of Turkey (T{\"U}B\.{I}TAK) Grant no:118F397.

\paragraph{Conflicts of interest/Competing interests}
The authors have no conflicts of interest to declare that are relevant to the content of this article.

\paragraph{Code availability}
The source codes are made available in public repositories, links to which are provided in the text and in the references.

\paragraph{Availability of data and material}
The extremal graphs output by our algorithms are made available in public repositories, links to which are provided in the text and in the references.

redpajama_set_name: RedPajamaArXiv


Look anywhere in the many halls of this year's IBC Show, and one of the most pressing issues being addressed in the future of live media viewing experiences – particularly in sports – was latency. That delay (or, ideally, lack thereof!) between delivery to various devices across the consumer landscape was the talk of the town, and few (if any) shown brighter under that microscope than Limelight Networks.
The company received numerous awards – including a IABM BaM Award and a TVBEurope Best of Show – for its new Limelight Realtime Streaming solution, which debuted for the first time in Amsterdam. According to Limelight, it's the industry's first globally scalable, sub-second live-video-streaming solution that leverages the industry-standard WebRTC format, which, in turn, helps natively support it in all traditional major web browsers.
Limelight Realtime Streaming is built around the idea of providing a service that can scale the delivery of WebRTC while still wrapping it around all of the features expected from a high-end CDN; things like multi-ingest, primary and backup ingest, streaming security and authentication, multi-bitrate and AVR output, geographic restrictions, and more.
It's that added layer of extremely low latency with the flexibility of the WebRTC format that Miller-Jones believes opens the door for exciting digital-sports experiences to come in the very near future.
The connection to sports betting, which is legal in most of Europe and was recently opened up for legalization across the United States, is a logical one. In fact, Limelight already has the sports data experts Sportradar on board as a client of the new service.
Already one of the top CDN providers in the world, Limelight Networks has truly positioned itself as a leader in the battle against latency over the past year. Back at the NAB Show, the company began pushing a new WebRTC-based live streaming solution that would grow to become Limelight Realtime Streaming. The company published an intriguing white paper tackling the issue of latency in real-time streaming over the summer. It's also already become a trusted partner of one of the biggest direct-to-consumer sports-streaming services, DAZN. Limelight solutions were utilized to broadcast, for example, J. League soccer from Japan to the rest of the globe.


San Francisco Review of Books: Interview: Wayne Allyn Root says "President Trump is a brilliant deal-maker, negotiator and salesman. Set him loose on China."
Interview: Wayne Allyn Root says "President Trump is a brilliant deal-maker, negotiator and salesman. Set him loose on China."
Editor's note: This article was originally published on January 31, 2017.
This is the second of five articles spanning my discussion with Wayne Allyn Root. The first piece can be read here. Material quoted below appeared in yesterday's article, offering background on Root and our discussion.
Joseph Ford Cotto: One reason the American economy fails to meet standards set by its postwar halcyon era is that it produces a decreasing number of material goods. What would you say could be done to reinvigorate our manufacturing sector?
Wayne Allyn Root: Simple. Dramatically cut the taxes and regulations. Make America among the lowest tax countries in the world. Kill all the job-killing regulations. Fill the government with businessmen like me who understand government is the enemy of economic growth and job creation. Get rid of bureaucrats and lawyers -- they are the job killers. Price out every regulation on the books currently, or proposed. If the cost is high and it destroys jobs, kill it, wipe it off the books. And for God's sake please kill Obamacare. Stick a sword through its heart. Stab it about 100 times to make sure it's dead.
Do all of that, and we will invent a new word for our success -- "Reverse Inversion." Every company in the world will want to move to America!
Cotto: China is notorious for its currency manipulation schemes. Beyond this, however, it not only owns a tremendous amount of America's national debt, but accounts for much of our trade deficit as well. How do you suppose that the U.S. could level the playing field in the near future?
Root: President Trump is a brilliant deal-maker, negotiator and salesman. Set him loose on China. Use the carrot and the stick. Ask them nicely to play ball and level the playing field…and offer a carrot they cannot refuse. Hint strongly that if they don't respond with a reasonable compromise, we will impose the stick. My bet- they will compromise, negotiate and level the playing field. But can you imagine how dumb Obama is -- in eight years, he never even asked. He never tried to level the playing field. If you don't ask, you don't get!


There are two main ways of searching in the SAGE Research Methods platform. The first is by using the quick search at the top of every page. The quick search performs a full text search for the terms you enter. A full text search finds every piece of content that contains the words you have entered. The items with the most occurrences of the terms will appear at the top of the list.
Following any search you can then apply filters to the results. These are on the right hand side of the search results page.
There are filters to show only what content is available to you, for content type and sub-type, for disciplines, a date range slide bar, and a filter to show whole works, chapters, entries or segments of videos.
If this doesn't get you what you are looking for you can try the second way of searching, an advanced search. The advanced search page can be accessed from underneath the quick search box at the top of every page in the SAGE Research Methods platform. On this page you can select and search many different elements within the platform. For more details on using this page see the Advanced Search Tips page.
Alternatively you can browse using the browse menu at the top of every page. You can browse by Topic, which will take you to the Methods Map, Discipline to see content relevant to a specific discipline, or by content type.


The Treehouse, at The Alnwick Garden, is hosting a Winter Wedding Fair on Monday (January 28).
It will take place from 5pm to 8pm and there will be music and experts on hand from the wedding industry, as well as the Treehouse wedding team.
Canapés and drinks will welcome you and everyone who attends will get a chance to win some prizes.
The venue can host up to 65 guests for wedding ceremonies and up to 85 for a reception.


Lavender is a wonderful smelling essential oil. When inhaled it helps reduce anxiety, stress, tension, headaches, panic attacks, and other central nervous system issues.
Lavender is very therapeutic. Its aroma is alluring and safe to wear in a work setting. Others will be attracted to the scent and it will help others relax too.


Previous Value: The Student College (StuCo) is being established to provide members of the Carnegie Mellon community with the opportunity to share in educational course-based experiences that are not available through regular university offerings, as led by Carnegie Mellon students. One of the most successful StuCo courses has been the Carnegie Mellon Community Garden course. The course has led to the robust development of two garden sites on the CMU campus both, managed by students. The garden team seeks to further expand the garden in the spring of 2013. Members of the garden team have engaged with the campus Children's School to provide garden education activities (Summer 2012) and have been active participants in other campus sustainability outreach programming, such as tabling during the annual Steinbrenner Environmental Expo.
StuCo is an educational organization involving hundreds of students at Carnegie Mellon University.
The Student College was established in 2001 to provide Carnegie Mellon students with the opportunity to share knowledge through educational, self-designed courses. Students can teach classes on any topic of their choice. However, the course cannot be available through regular University offerings. Instructors and students receive credit for their work.
•All currently-enrolled CMU students, staff and community members are eligible to take StuCo courses.
Previous Value: Courses typically meet once a week (for a full semester) and follow the current Carnegie Mellon academic calendar.
Previous Value: .CMU students are eligible to teach StuCo courses and to join the Executive Committee that governs StuCo. The StuCo Executive Committee is comprised of the Officers and Members At Large. StuCo is governed by this committee along with faculty advisors. Student teachers are not compensated for teaching the courses.
New Orientation Eco-Advisors, Peer Health Advisors, House Council Based students, StuCo, Green Practices Interns, Graduate Students; Net Impact, CEE Service Projects.


text:
The First Doctor Adventures Volume 4
By: Barnes, Jonathan, Smith, Andrew, Carter, Howard(By (composer)), Webster, Tom(Cover design or artwork by), Bentley, Ken(Director), Bradley, David(Performed by), Glover, Jamie(Performed by), Grant, Claudia(Performed by), Powell, Jemma(Performed by)
Part of the Doctor Who The First Doctor Adventures series
Big Finish Productions Ltd
This release features a recreation of the first ever cast of Doctor Who, as seen on BBC TV in 2013's celebratory An Adventure in Space and Time, with David Bradley then appearing as the Doctor in 2017's Peter Capaldi finale.
Contains two new adventures: 4.1 Return to Skaro by Andrew Smith. A new plan to return to Earth actually returns the TARDIS to another place its crew have recently visited - Skaro, home world of the Daleks.
But it is some time after their previous visit. The Thals have moved on with developing their species - Yet the shadow of the Dalek city always looms large over them.
Venturing into the abandoned metropolis, the Doctor and his friends discover the Daleks aren't as dead as they might have thought - and it isn't only their enemies who have secrets. 4.2 Last of the Romanovs by Jonathan Barnes. The TARDIS lands on Earth near to an eerie and familiar house... with the only witness a regal man watching from inside through a broken window. Leaving the ship, the crew immediately find themselves in trouble - because they have landed in Ekaterinburg early in the twentieth century. The man inside the house is Nicholas, the last Tsar of Russia, imprisoned with his family...and one of the most notorious crimes in history is just about to happen.
CAST: David Bradley (The Doctor), Claudia Grant (Susan), Jemma Powell (Barbara Wright), Jamie Glover (Ian Chesterton), Nicholas Briggs (The Daleks), John Albasiny (Makerenkov), James Camp (Jyden), Nigel Hastings (Kreos), Leighton Pugh (Nicholas Alexandrovich Romano/Thomas Preston), Alisdair Simpson (Damadus), Dan Starkey (Yakov Yurovsky), Alex Tregear (Anastasia Romanov), George Weightman (Grigory Nikulin), Tracy Wiles (Tryana).
FLS Space opera

redpajama_set_name: RedPajamaCommonCrawl


In 2015, the big four U.S marketplaces, namely Amazon, eBay, Etsy and Jet, sold a total of over $100 billion worth of goods. And there's no sign that these numbers will stop going upwards in the near future. According to the U.S Department of Commerce, online retail sales nearly quadrupled over the 10-year period from 2005 to 2015. Furthermore, this figure is projected to reach around $500 billion in 2018.
Traditional retailers have started to feel the impact of online marketplaces by making certain changes to adopt the growing trend of online shopping. Recently, Walmart has acquired Jet.com, tapping into the e-commerce competition, as they realized purchasing habits are gradually shifting from physical stores to online marketplaces.
That online e-commerce marketplace is rapidly booming and expected to become the new shopping destination can be broken down into multiple advantages it has over brick-and-mortar retailers. Millions of people shop at Amazon or eBay because of the extreme comfort these e-commerce platforms provide. With just a few clicks, while staying at home, consumers conveniently shop around the website, browse through a vast selection of items, read reviews from other customers, and compare prices before making the final buying decision.
WordPress is a powerful content management system (CMS) that can cater to any type of business you intend to build. Doing a small research about online business currently using WordPress, you'll be amazed at its popularity among different types of websites. Creating a freelance website is hence totally a thing you can do in WordPress. This article will show important features and guide to set up freelance website with WordPress.
Big freelance websites like Upwork, Freelancer, and Guru are doing an excellent job in connecting people in need of projects done with millions of service providers. One of the core values contributing to the success of such marketplaces is a stable platform with all functions operate seamlessly. Therefore, knowing how to setup your site properly is very important.
In a freelance marketplace, project bidding function is used as a typical approach. To discuss in depth the project bidding on a freelance site, this article will go through this function's definition, the reasons why it's widely employed, and the workflow of how freelancer bids work on your website.
Freelancing industry is quickly emerging, excellently leveraging freelancer workforce all around the world. With technology being the major driving force, digital services have made profound impact on the working trend and interested a great number of people who prefer to work as freelancers. Building a freelance marketplace connecting freelancers and clients promises potential growth and success; still, it comes with challenges of a proper freelance business model and the main concept that makes your site run smoothly. If business model cares about how to monetize your platform, the main concept is the central workflow your users follow to accomplish the transaction.
"Make money online" as a freelancer has increasingly been a popular choice among young people. To build a platform connecting freelancers and clients, hence, has great potential to grow. If you plan to start your business career following this path, one of the first things to take into consideration is the proper freelance business model you will use for your business.
For an online freelance site, which business models available and how each differs from the others? Generally speaking, business model is the strategy on what you will do with your site to generate revenue. A good business idea together with an appropriate business model promises to be an effective monetization tool.
Freelancing websites play a perfect role in connecting clients in need of getting projects done and freelancers, from all over the world, who are willing to provide services and get paid. For two sides to come together, freelancers can bid for any potential projects and clients will pick the most suitable one for their projects.
Search box is among those elements that are usually underestimated and put behind big things like sales pages. However, an advanced search box is an effective navigation tool giving users quick access to information, which saves users a lot of time and makes them feel happier. Therefore, designing a search bar on your website is necessary to create a smoother user flow.
In the previous article, I went over important notes of Design Principle 1 – how to create a good user flow for service providers. This post will cover the other 3 principles – what you need to focus on to provide buyers with an ideal micro job marketplace destination, and how to design account and profile settings.
Theoretically, a micro job site design must include 4 main principles: buying, selling, profile settings, and account settings. According to your business model, these four principles are to be considered thoroughly to match the concept of your platform. That said, among these 4 fundamentals, selling and buying activities are the most important as they are responsible for triggering transactions between buyers and sellers on your micro job marketplace. Meanwhile, transactions are the ultimate purpose you aim to capture. Hence, a strong platform is more likely to handle micro job orders seamlessly.
This article will discuss in depth the first principle in a series of 4 principles to provide you with an overview about how to design your micro job marketplace and make it work effectively.
Outsourcing service has become popular thanks to multiple benefits it brings to business. It is especially the case for small and medium businesses since you can't do all the jobs by yourself. Freelance websites have raised as a platform connecting project owners and skillful freelancers.


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Most parents are familiar with the idea of visiting amusement parks or campsites to spend their family holidays. With those two places being the most prominent destinations, people tend to perceive it as the places that offer more values than the other sites. However, there are still other options that parents can choose to have fun with their family, especially when they have little kids around the age of seven to ten. These less mainstream choices include visiting historical sites.
At this point, one may wonder about the benefits of visiting such places. There are indeed advantages that both parents and children can benefit from the activity, and it relates to the concept of recreational education, quite similar to what teachers do when they take the students to the museums. Compared to another mainstream holiday idea, visiting sites full of Arca and relics is indeed less popular. However, it still can be a great choice for those smart parents.
Parents may not expect their children to get a perfect score in the history lesson. However, taking them to the sites will increase their understanding of what happened in those specific areas. Of course, a parental guide is very much necessary to explain things and to prevent the children from grasping the wrong ideas about the records. Thus, it is also an excellent choice if you seek to strengthen the family bonds between you and your children.
Old people's wise sayings always highlight the importance of learning from history. Although the phrase holds a lot more definitions that what most people comprehend, it is still a great starter to teach your children about life lessons. Holocaust memorials, for instance, are indeed a perfect spot to tell your children about what happened in the past and what they could have done to stop the massacre. Remember to always present the narration in a light tone as a way not to scare the kids. As plus points, the kids will eventually learn to be better people, and you will earn their respects.


text:
Manic depression and creativity, Kay Jamison
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Interviewee: Kay Jamison. More than 20 scientific studies show an elevated rate of manic depression among highly creative people. Here Kay Jamison discusses the highs and lows of manic depression, the link between mania and creativity, and the heritability of the disorder. (DNAi Location: Chronicle > Living with eugenics > Living with depression > Manic depression and creativity)
One thing that is clear is that if you look at the scientific literature there are more than 20 studies showing a very much elevated rate of depression and manic depression in people who are highly creative. Obviously most people who are highly creative are, don't suffer from any kind of psychopathology, they aren't particularly moody, they are, don't suffer from depression, it's rather that there is a disproportionate rate. And I think that when you're talking about genius, when you're talking about someone with the mind of someone like Byron, for example, or Virginia Woolf, I think you're talking about people who are by definition, it's a very unusual set of circumstances to, that leads to that kind of mind and ability. And in both of those instances a very strong family history, it's a genetic illness, a very strong family history of depression and mania and suicide. And in both of those instances you could see how these illnesses has both gave them a range of experience from which they wrote, but it also gave an intensity and a power and a fluency to that which they wrote.
manic depression and creativity,kay jamison,depression and creativity,byron virginia,highs and lows,history of depression,genetic illness,disproportionate rate,dnai,heritability,strong family,interviewee,virginia woolf,psychopathology,fluency,family history,illnesses,instances,intensity,genius
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1061. Bipolar Disorder and Creativity
Kay Jamison discusses the link between mania and creativity, and the heritability of the disorder.
Source: G2C
15375. Manic depression as a genetic disorder, Kay Jamison
Kay Jamison discusses the highs and lows of manic depression, the link between mania and creativity, and the heritability of the disorder.
15371. Manic depression and the thought of suicide, Kay Jamison
15370. Understanding the experience of a manic depressive, Kay Jamison
15372. Manic depression and mania, Kay Jamison
1996. Bipolar disorder and creativity (2)
Professor James Potash discusses evidence from a number of studies that individuals with mood disorders are more likely to be highly creative.
1056. Experiencing Bipolar Disorder
Kay Jamison discusses the highs and lows of manic depression (bipolar disorder), the link between mania and creativity, and the heritability of the disorder.
15376. Drawing family pedigrees for manic depression, Kay Jamison
Kay Jamison discusses her family pedigree, which is similar to the Kallikak pedigree.
15374. The maternal side of her family pedigree, Kay Jamison
15380. Manic deprssion as a destructive yet treatable illness, Kay Jamison
Kay Jamison describes the shock of hearing her own diagnosis of manic depression and her doctor's advice not to have children.