Size-Ramsey numbers of graphs with maximum degree three

The size-Ramsey number $\widehat{r}(H)$ 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 $\widehat{r}(H)=O(n^{8\mathrm{/}5})$, improving upon the upper bound of $n^{5\mathrm{/}3+o(1)}$ by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper we show that $\widehat{r}(H)\le n^{3\mathrm{/}2+o(1)}$. 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.