Related papers: Ramsey numbers of grid graphs

The odd-Ramsey number $r_{{\text odd}}(n,H)$ of a graph $H$, as introduced by Alon in his work on graph-codes, is the minimum number of colours needed to edge-colour $K_n$ so that every copy of $H$ intersects some colour class in an odd…

Given a nonempty graph $G$, a collection of nonempty graphs $\cal{H}$, and a positive integer $k$, the Gallai-Ramsey number $\mathrm{gr}_k(G:\mathcal{H})$ is defined to be the minimum positive integer $n$ such that every exact…

For given simple graphs $H_1,H_2,\dots,H_c$, the multicolor Ramsey number $R(H_1,H_2,\dots,H_c)$ is defined as the smallest positive integer $n$ such that for an arbitrary edge-decomposition $\{G_i\}^c_{i=1}$ of the complete graph $K_n$, at…

Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle…

For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $ H $…

For two graphs $G,H$ and a positive integer $k$, the \emph{Gallai-Ramsey number} $\operatorname{gr}_k(G,H)$ is defined as the minimum number of vertices $n$ such that any $k$-edge-coloring of $K_n$ contains either a rainbow (all different…

Given bipartite graphs $H_1$, \dots , $H_k$, the bipartite Ramsey number $br(H_1,\dots, H_k)$ is the minimum integer $N$ such that any $k$-edge-coloring of complete bipartite graph $K_{N, N}$ contains a monochromatic $H_i$ in color $i$ for…

Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $\chi(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph…

Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In…

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…

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…

Let $G$, $H$ and $K$ represent three graphs without loops or parallel edges and $n$ represent an integer. Given any red blue coloring of the edges of $G$, we say that $K \rightarrow (G,H)$, if there exists red copy of $G$ in $K$ or a blue…

Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the…

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…

Given two graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ denotes the smallest integer $N$ such that any red-blue coloring of the edges of $K_N$ contains either a red $G_1$ or a blue $G_2$. Let $G_1$ be a graph with chromatic number…

Let $G$ and $H$ be finite graphs. If for any two-coloring of the edges of a complete graph $K_n$, there is a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue, we will say $K_n\rightarrow (G,H)$. The Ramsey…

Let $G_1, G_2, ..., G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, ..., G_t)$ is the smallest positive integer $n$ such that if the edges of complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs…

Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in…

Given a graph $H$, let $\chi_H(\mathbb{R}^n)$ be the smallest positive integer $r$ such that there exists an $r$-coloring of $\mathbb{R}^n$ with no monochromatic unit-copy of $H$, that is a set of $|V(H)|$ vertices of the same color such…

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the…