Bipartite subgraphs

AbstractIt is shown that there exists a positivec so that for any large integerm, any graph with 2m2edges contains a bipartite subgraph with at least $$m^2 + m/2 + c\sqrt m$$ edges. This is tight up to the constantc and settles a problem of Erdös. It is also proved that any triangle-free graph withe>1 edges contains a bipartite subgraph with at least e/2+c′ e4/5 edges for some absolute positive constantc′. This is tight up to the constantc′.

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