Related papers: Small Ramsey numbers for books, wheels, and genera…

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set…

We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$…

The induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is defined as the minimum order of a graph $F$ on such that any 2-coloring of its edges with red and blue leads to either a red induced copy of $G$ or a blue induced copy of $H$. Motivated…

Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\{v_0,v_1,\ldots,v_{n_1},w_0,w_1,\ldots,w_{n_2}\}$ and…

In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $\bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs…

Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a…

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it…

For graphs $G_1, G_2, G_3$, the three-color Ramsey number $R(G_1,$ $G_2, G_3)$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with 3 colors, then it contains a monochromatic copy…

Given a graph $H$ and a positive integer $k$, the {\it $k$-colored Ramsey number} $R_k(H)$ is the minimum integer $n$ such that in every $k$-edge-coloring of the complete graph $K_{n}$, there is a monochromatic copy of $H$. Given two graphs…

We created and parallelized two SAT solvers to find new bounds on some Ramsey-type numbers. For $c > 0$, let $R_c(L)$ be the least $n$ such that for all $c$-colorings of the $[n]\times [n]$ lattice grid there will exist a monochromatic…

Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a…

We investigate Ramsey properties of a random graph model in which random edges are added to a given dense graph. Specifically, we determine lower and upper bounds on the function $p=p(n)$ that ensures that for any dense graph $G_n$ a.a.s.…

Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either a…

Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide…

An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. Given a positive integer $n$ and a graph $G$, the \textit{anti-Ramsey number} $ar(n,G)$ is defined to be the minimum number of colors $r$…

This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number $R(4,3,3)=30$. The number…

The anti-Ramsey number, $AR(n,G)$, for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy…

The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. The Ramsey numbers of $B_n^{(k)}$ are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author…

The Ramsey number $R(s,t)$ is the smallest integer $n$ such that all graphs of size $n$ contain a clique of size $s$ or an independent set of size $t$. $\mathcal{R}(s,t,n)$ is the set of all counterexample graphs without this property for a…

The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest number of edges a graph $G$ can have, such that for every edge-coloring of $G$ with $r$ colors there exists a monochromatic copy of $H$ in $G$. For a graph $H$, we…