  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 $t(G) \geq b(G)$ 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>C n^{-1/2}
\log^{1/2}n $ for a suitable constant $C$, which is best possible up to the
value of $C$.

Probability;Discrete Mathematics;Combinatorics