  For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a
graph $G$ has the property that all subsets of size $\alpha n$ contain the
``correct'' number of copies of $H$ one would expect to find in the random
graph $G(n,p)$ then $G$ behaves like the random graph $G(n,p)$; that is, it is
$p$-quasi-random in the sense of Chung, Graham, and Wilson. This solves a
conjecture raised by Shapira and solves in a strong sense an open problem of
Simonovits and S\'os.
