The size-Ramsey number of cubic graphs

We show that the size-Ramsey number of any cubic graph with $n$ vertices is $O(n^{8/5})$, improving a bound of $n^{5/3 + o(1)}$ due to Kohayakawa, R\"{o}dl, Schacht, and Szemer\'{e}di. The heart of the argument is to show that there is a constant $C$ such that a random graph with $C n$ vertices where every edge is chosen independently with probability $p \geq C n^{-2/5}$ is with high probability Ramsey for any cubic graph with $n$ vertices. This latter result is best possible up to the constant.