The size‐Ramsey number of cubic graphs

We show that the size-Ramsey number of any cubic graph with n n vertices is O ( n 8 / 5 ) O(n8/5) , improving a bound of n 5 / 3 + o ( 1 ) n5/3 + o(1) due to Kohayakawa, Rödl, Schacht, and Szemerédi. The heart of the argument is to show that there is a constant C C such that a random graph with C n $C n$ vertices where every edge is chosen independently with probability p ⩾ C n − 2 / 5 p C n-2/5 is with high probability Ramsey for any cubic graph with n n vertices. This latter result is best possible up to the constant.