Tight Ramsey bounds for multiple copies of a graph

TL;DR

This paper determines the exact threshold number of copies needed for the Ramsey number of multiple disjoint copies of a graph to follow a known linear formula, significantly improving previous bounds.

Contribution

The authors establish that the asymptotic behavior of the Ramsey number for multiple copies of a graph occurs at a single exponential number of copies, refining longstanding bounds.

Findings

Long-term behavior occurs at a single exponential number of copies

Improves previous triple exponential bounds

Provides an essentially tight answer to a 30-year-old problem

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, Erd\H{o}s and Spencer in 1975, who showed $r(nH)=(2|H|-\alpha(H))n+c$, for some $c=c(H)$, provided $n$ is large enough. Since it did not follow from their arguments, Burr, Erd\H{o}s 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, Erd\H{o}s and Spencer by showing that the long term behaviour occurs already when the number of copies is single exponential.