Related papers: Ramsey-type problems for tilings in dense graphs

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…

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…

Given two graphs $G$ and $H$, the {Ramsey number} $R(G,H)$ is the smallest positive integer $N$ such that every 2-coloring of the edges of $K_{N}$ contains either a red $G$ or a blue $H$. Let $K_{N-1}\sqcup K_{1,k}$ be the graph obtained…

For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$…

Over recent years there has been much interest in both Tur\'an and Ramsey properties of vertex ordered graphs. In this paper we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a…

Burr and Erd\H{o}s in 1975 conjectured, and Chv\'atal, R\"odl, Szemer\'edi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed…

We prove that $s_r(K_{k+1}) = O(k^3 r^3 \log^3 k)$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erd\H{o}s and Lov\'{a}sz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of…

Given a graph $H$, the $k$-colored Gallai Ramsey number $gr_{k}(K_{3} : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a…

A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_<(H)$ is the minimum $n$ such that every edge-2-coloring of the…

Given simple graphs $H_{1},H_{2},\ldots,H_{c}$, the Ramsey number $r(H_{1},H_{2},\ldots,H_{c})$ is the smallest positive integer $n$ such that every edge-colored $K_{n}$ with $c$ colors contains a subgraph in color $i$ isomorphic to $H_{i}$…

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$…

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…

In a $(G^1,G^2)$ coloring of a graph $G$, every edge of $G$ is in $G^1$ or $G^2$. For two bipartite graphs $H_1$ and $H_2$, the bipartite Ramsey number $BR(H_1, H_2)$ is the least integer $b\geq 1$, such that for every $(G^1, G^2)$ coloring…

The multicolor Ramsey number $r_k(F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_n$ by $k$ colors there is a monochromatic copy of $F$. In this short note we prove an upper bound on $r_k(F)$ for a…

The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In…

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…

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete…

For a positive integer $k$ and a graph $H$, the $k$-color induced size-Ramsey number $\hat{R}_{\mathrm{ind}}(H, k)$ is the minimum integer $m$ for which there exists a graph $G$ with $m$ edges such that for every $k$-edge coloring of $G$,…

Given a labeled graph $H$ with vertex set $\{1, 2,\ldots,n\}$, the ordered Ramsey number $r_<(H)$ is the minimum $N$ such that every two-coloring of the edges of the complete graph on $\{1, 2, \ldots,N\}$ contains a copy of $H$ with…

A graph $G$ is $q$-Ramsey for another graph $H$ if in any $q$-edge-colouring of $G$ there is a monochromatic copy of $H$, and the classic Ramsey problem asks for the minimum number of vertices in such a graph. This was broadened in the…