Related papers: List Ramsey numbers

We prove that if $G$ is a sparse graph --- it belongs to a fixed class of bounded expansion $\mathcal{C}$ --- and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these…

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

An edge-ordered graph is a graph with a linear ordering of its edges. Two edge-ordered graphs are equivalent if their is an isomorphism between them preserving the ordering of the edges. The edge-ordered Ramsey number $r_{edge}(H; q)$ of an…

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this…

We improve the upper bound for diagonal Ramsey numbers to \[R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}\] for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended…

We show that, for $n$ large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic $K_t$s in any two-colouring of the edges of $K_n$, where $C \approx 2.18$ is an explicitly defined constant. The old lower bound, due to…

A graph is $k$-clique-extendible if there is an ordering of the vertices such that whenever two $k$-sized overlapping cliques $A$ and $B$ have $k-1$ common vertices, and these common vertices appear between the two vertices $a,b\in…

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…

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 $K_{N,N}$…

We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Soki\'c. As a bonus, our proof…

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has…

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…

A graph $H$ is common if its Ramsey multiplicity, i.e., the minimum number of monochromatic copies of $H$ contained in any $2$-edge-coloring of $K_n$, is asymptotically the same as the number of monochromatic copies in the random…

We study a generalization of a famous result of Goodman and establish that asymptotically at least a $1/256$ fraction of all triangles needs to be monochromatic in any four-coloring of the edges of a complete graph. We also show that any…

For an $r$-graph $H$, the anti-Ramsey number ${\rm ar}(n,r,H)$ is the minimum number $c$ of colors such that for any edge-coloring of the complete $r$-graph on $n$ vertices with at least $c$ colors, there is a copy of $H$ whose edges have…

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…

A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are…

We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we…

In [5] Graham and Rothschild consider a geometric Ramsey problem: finding the least n such that if all edges of the complete graph on the points {+1,-1}^n are 2-colored, there exist 4 coplanar points such that the 6 edges between them are…

In 2015 Bloom and Liebenau proved that $K_n$ and $K_n+K_{n-1}$ possess the same $2$-Ramsey graphs for all $n\geq 3$ (with a single exception for $n=3$). In the following we give a simple proof that $K_n$ and $K_n+K_{n-1}$ possess the same…