Extremal subgraphs of random graphs

We shall prove that if L is a 3‐chromatic (so called “forbidden”) graph, and — R n is a random graph on n vertices, whose edges are chosen independently, with probability p , and — B n is a bipartite subgraph of R n of maximum size, — F n is an L ‐free subgraph of R n of maximum size, then (in some sense) F n and B n are very near to each other: almost surely they have almost the same number of edges, and one can delete O p (1) edges from F n to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, F n is almost surely bipartite.