Related papers: Minimum degrees and codegrees of minimal Ramsey 3-…

For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $\Phi :E(G)\longrightarrow E( \mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq \Phi (e)$. We denote by $\mathcal{B}^rG$, the…

For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if every hyperedge colouring with $k$ colours of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous…

For a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $G$ (or a Berge-$G$ in short), if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We denote the family of…

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be…

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…

Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In previous work, we proved the same result for 3-uniform hypergraphs. Here we extend this result to…

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we…

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 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…

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…

Let $C_6^3$ be the 3-uniform hypergraph on $\{1,\dots, 6\}$ with edges $123, 345,561$, which can be seen as the triangle in 3-uniform hypergraphs. For sufficiently large $n$ divisible by 6, we show that every $n$-vertex 3-uniform hypergraph…

Let $R_h(k; \ell)$ be the smallest integer $n$ such that any edge coloring of a complete graph on $n$ vertices in $\ell$ colors results in a monochromatic $K_k$-minor, in other words, a graph with Hadwiger number $k$, i.e., a graph that…

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…

Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a…

Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey…

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…

An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$…

We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$…

Motivated by Ramsey theory problems, we consider edge-colorings of 3-uniform hypergraphs that contain no rainbow paths of length 3. There are three 3-uniform paths of length 3: the tight path $\mathcal{T}=\{v_1v_2v_3, v_2v_3v_4,…

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…