Size‐Ramsey numbers of graphs with maximum degree three

Abstract The size‐Ramsey number of a graph is the smallest number of edges a (host) graph can have, such that for any red/blue colouring of , there is a monochromatic copy of in . Recently, Conlon, Nenadov and Trujić showed that if is a graph on vertices and maximum degree three, then , improving upon the upper bound of by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper, we show that . While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.