Ramsey numbers for multiple copies of sparse graphs

Abstract

AbstractFor a graph and an integer , we let denote the disjoint union of copies of . In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for , one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant such that , provided is sufficiently large. Subsequently, Burr gave an implicit way of computing and noted that this long‐term behaviour occurs when is triply exponential in . Very recently, Bucić and Sudakov revived the problem and established an essentially tight bound on by showing follows this behaviour already when the number of copies is just a single exponential. We provide significantly stronger bounds on in case is a sparse graph, most notably of bounded maximum degree. These are relatable to the current state‐of‐the‐art bounds on 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.