Ramsey numbers for multiple copies of sparse graphs

Abstract

For a graph $H$ and an integer $n$, we let $nH$ denote the disjoint union of $n$ copies of $H$. In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for $nH$, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant $c=c\left(H\right)$ such that $r\left(nH\right)=\left(2\mid H\mid -\alpha \left(H\right)\right)n+c$, provided $n$ is sufficiently large. Subsequently, Burr gave an implicit way of computing $c$ and noted that this long-term behaviour occurs when $n$ is triply exponential in $\mid H\mid$. Very recently, Bucić and Sudakov revived the problem and established an essentially tight bound on $n$ by showing $r\left(nH\right)$ follows this behaviour already when the number of copies is just a single exponential. We provide significantly stronger bounds on $n$ in case $H$ is a sparse graph, most notably of bounded maximum degree. These are relatable to the current state-of-the-art bounds on $r\left(H\right)$ and (in a way) tight. Our methods rely on a beautiful classic proof of Graham, Rödl and Ruciński, with an emphasis on developing an efficient absorbing method for bounded degree graphs.