Extremal subgraphs of random graphs

Let K` denote the complete graph on ` vertices. We prove that there is a constant c = c(`), such that whenever p ≥ n−c, with probability tending to 1 when n goes to infinity, every maximum K`-free subgraph of the binomial random graph Gn,p is (`− 1)partite. This answers a question of Babai, Simonovits and Spencer [BSS90]. The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with M edges, where M n, is nearly unique. More precisely, given a maximum cut C of Gn,M , we can obtain all maximum cuts by moving at most O( √ n3/M) vertices between the parts of C.