Ramsey numbers for multiple copies of sparse graphs

TL;DR

This paper improves bounds on the Ramsey numbers for multiple copies of sparse graphs, especially those with bounded maximum degree, using advanced absorbing methods to refine long-standing theoretical results.

Contribution

It provides significantly stronger bounds on the number of copies needed for the Ramsey number behavior in sparse graphs, extending prior results with new methods.

Findings

Stronger bounds for Ramsey numbers of multiple sparse graphs.

Applicable to graphs with bounded maximum degree.

Methodology based on an efficient absorbing technique.

Abstract

For a graph $H$ and an integer $n$, we let $nH$ denote the disjoint union of $n$ copies of $H$. In 1975, Burr, Erd\H{o}s, 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 $c = c(H)$ such that $r(nH) = (2|H| - \alpha(H))n + c$, 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 $|H|$. Very recently, Buci\'{c} and Sudakov revived the problem and established an essentially tight bound on $n$ by showing $r(nH)$ 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 $r(H)$ and (in a way) tight. Our methods rely on a beautiful classic proof of Graham, R\"{o}dl, and Ruci\'{n}ski, with the emphasis on developing an efficient absorbing method for bounded degree graphs.