  We prove that for every non-trivial hereditary family of graphs ${\cal P}$
and for every fixed $p \in (0,1)$, the maximum possible number of edges in a
subgraph of the random graph $G(n,p)$ which belongs to ${\cal P}$ is, with high
probability, $$ \left(1-\frac{1}{k-1}+o(1)\right)p{n \choose 2}, $$ where $k$
is the minimum chromatic number of a graph that does not belong to ${\cal P}$.
