Subgraphs of random graphs in hereditary families

For a graph G and a hereditary property P, let ex (G, P) denote the maximum number of edges of a subgraph of G that belongs to P. We prove that for every non-trivial hereditary property P such that L P for some bipartite graph L and for every fixed p (0, 1) we have (G (n, p), P) n^2-\ with high probability, for some constant = (P) >0. This answers a question of Alon, Krivelevich and Samotij.