  In 1977, Erd\H{o}s and Hajnal made the conjecture that, for every graph $H$,
there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable
set of size at least $|G|^c$; and they proved that this is true with $ |G|^c$
replaced by $2^{c\sqrt{\log |G|}}$. Until now, there has been no improvement on
this result (for general $H$).
  We prove a strengthening: that for every graph $H$, there exists $c>0$ such
that every $H$-free graph $G$ with $|G|\ge 2$ has a clique or stable set of
size at least $$2^{c\sqrt{\log |G|\log\log|G|}}.$$ Indeed, we prove the
corresponding strengthening of a theorem of Fox and Sudakov, which in turn was
a common strengthening of theorems of R\"odl, Nikiforov, and the theorem of
Erd\H{o}s and Hajnal mentioned above.
