Related papers: On some multicolour Ramsey properties of random gr…

Given a pair of $k$-uniform hypergraphs $(G,H)$, the Ramsey number of $(G,H)$, denoted by $R(G,H)$, is the smallest integer $n$ such that in every red/blue-colouring of the edges of $K_n^{(k)}$ there exists a red copy of $G$ or a blue copy…

We consider extremal edge-coloring problems inspired by the theory of anti-Ramsey / rainbow coloring, and further by odd-colorings and conflict-free colorings. Let $G$ be a graph, and $F$ any given family of graphs. For every integer $n…

Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of…

A well-known result of R\"odl and Ruci\'nski states that for any graph $H$ there exists a constant $C$ such that if $p \geq C n^{- 1/m_2(H)}$, then the random graph $G_{n,p}$ is a.a.s. $H$-Ramsey, that is, any $2$-colouring of its edges…

In an $r$-coloring of edges of the complete graph on $n$ vertices, how many edges are there in the largest monochromatic connected component? A construction of Gy\'arf\'as shows that for infinitely many values of $r$, there exist colorings…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed…

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of…

For simple graphs $G$ and $H$, their size Ramsey number $\hat{r}(G,H)$ is the smallest possible size of $F$ such that for any red-blue coloring of its edges, $F$ contains either a red $G$ or a blue $H$. Similarly, we can define the…

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 $…

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…

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…

Given two graphs $H_1$ and $H_2$, an online Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round Builder selects an edge and Painter colors it red or blue. Builder is trying to force Painter to create a red copy of $H_1$…

Given a positive integer $s$, the $s$-colour size-Ramsey number of a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that, in any colouring of $E(G)$ with $s$ colours, there is a…

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…

Given a positive integer $ r $, the $ r $-color size-Ramsey number of a graph $ H $, denoted by $ \hat{R}(H, r) $, is the smallest integer $ m $ for which there exists a graph $ G $ with $ m $ edges such that, in any edge coloring of $ G $…

For edge-ordered graphs $G^{\prec}$ and $H^{\prec}$, the size edge-ordered Ramsey number $\hat{r}_{\text{edge}}(G^{\prec}, H^{\prec})$ is defined as the smallest integer $m$ for which there exists an edge-ordered graph $F^{\prec}$ (with…

Given a graph $H$, the Ramsey number $r(H)$ is the smallest natural number $N$ such that any two-colouring of the edges of $K_N$ contains a monochromatic copy of $H$. The existence of these numbers has been known since 1930 but their…

The $r$-color size-Ramsey number of a graph $H$, denoted by $\widehat{R}_r(H)$, is the minimum number of edges in a graph $G$ having the property that every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H$. Krivelevich…