Induced Subgraph Density. I. A loglog Step Towards Erd̋s–Hajnal

In 1977, Erd̋s 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 $$ \begin{align*} &2^{c\sqrt{\log |G|\log\log|G|}}.\end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above.