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Extremal subgraphs of random graphs

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We prove that there is a constant c > 0, such that whenever p ≥ n -c, with probability tending to 1 when n goes to infinity, every maximum triangle-free subgraph of the random graph G n,p is bipartite. This answers a question of Babai, Simonovits and Spencer (Babai et al., J Graph Theory 14 (1990) 599-622). 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 and M ≤ (n 2)/2, is "nearly unique". More precisely, given a maximum cut C of G n,M, we can obtain all maximum cuts by moving at most O(√n 3/M) vertices between the parts of C.

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