  We confirm a conjecture of Fox, Pach, and Suk, that for every $d>0$, there
exists $c>0$ such that every $n$-vertex graph of VC-dimension at most $d$ has a
clique or stable set of size at least $n^c$. This implies that, in the language
of model theory, every graph definable in NIP structures has a clique or
anti-clique of polynomial size, settling a conjecture of Chernikov, Starchenko,
and Thomas.
  Our result also implies that every two-colourable tournament satisfies the
tournament version of the Erd\H{o}s-Hajnal conjecture, which completes the
verification of the conjecture for six-vertex tournaments. The result extends
to uniform hypergraphs of bounded VC-dimension as well.
  The proof method uses the ultra-strong regularity lemma for graphs of bounded
VC-dimension proved by Lov\'asz and Szegedy and the method of iterative
sparsification introduced by the authors in an earlier paper.
