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Let K_l denote the complete graph on l vertices. We prove that there is a constant c
= c(l) > 0, such that whenever p >= n^{-c}, with probability tending to 1 when n goes
to infinity, every maximum K_l-free subgraph of the binomial random graph G_{n, p}
is (l-1)-partite. This answers a question of Babai, Simonovits and Spencer.
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 G_{n, M}, we can obtain all maximum cuts by
moving at most O (n^{3/2}/M) vertices between the parts of C.
Joint work with Graham Brightwell and Angelika Steger.
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