  The Erd\H{o}s-Hajnal conjecture says 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 $2^{c\log|G|}$ (a graph is ``$H$-free'' if no induced subgraph is
isomorphic to $H$). The conjecture is known when $H$ is a path with at most
four vertices, but remains open for longer paths. For the five-vertex path,
Blanco and Buci\'c recently proved a bound of $2^{c(\log |G|)^{2/3}}$; for
longer paths, the best existing bound is $2^{c(\log|G|\log\log|G|)^{1/2}}$.
  We prove a much stronger result: for any path $P$, every $P$-free graph $G$
has a clique or stable set of size at least $2^{(\log |G|)^{1-o(1)}}$. We
strengthen this further, weakening the hypothesis that $G$ is $P$-free by a
hypothesis that $G$ does not contain ``many'' copies of $P$, and strengthening
the conclusion, replacing the large clique or stable set outcome with a
``near-polynomial'' version of Nikiforov's theorem.
