Tight Ramsey bounds for multiple copies of a graph

Abstract

The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs $G$ for which the value of $r(G)$ is known exactly. One such family consists of large vertex disjoint unions of a fixed graph $H$, we denote such a graph, consisting of $n$ copies of $H$ by $nH$. This classical result was proved by Burr, Erdős and Spencer in 1975, who showed $r(nH)=(2|H|-α(H))n+c$, for some $c=c(H)$, provided $n$ is large enough. Since it did not follow from their arguments, Burr, Erdős and Spencer further asked to determine the number of copies we need to take in order to see this long term behaviour and the value of $c$. More than $30$ years ago Burr gave a way of determining $c(H)$, which only applies when the number of copies $n$ is triple exponential in $|H|$. In this paper we give an essentially tight answer to this very old problem of Burr, Erdős and Spencer by showing that the long term behaviour occurs already when the number of copies is single exponential.