Size-Ramsey numbers of graphs with maximum degree three

Abstract

The size-Ramsey number $\hat{r}\left(H\right)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Trujić showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}\left(H\right)=O\left(n^{8/5}\right)$, improving upon the upper bound of $n^{5/3+o\left(1\right)}$ by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper, we show that $\hat{r}\left(H\right)⩽n^{3/2+o\left(1\right)}$. 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.