For a graph G, denote by t(G) (resp. b(G)) the maximum size of a
triangle-free (resp. bipartite) subgraph of G. Of course t(G)≥b(G) for
any G, and a classic result of Mantel from 1907 (the first case of Tur\'an's
Theorem) says that equality holds for complete graphs. A natural question,
first considered by Babai, Simonovits and Spencer about 20 years ago is, when
(i.e. for what p=p(n)) is the "Erd\H{o}s-R\'enyi" random graph G=G(n,p)
likely to satisfy t(G)=b(G)? We show that this is true if p>Cn−1/2log1/2n for a suitable constant C, which is best possible up to the
value of C.Comment: 15 page