A loglog step towards the Erdős-Hajnal conjecture
Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.
Abstract
In 1977, Erdős 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 recently proved 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 $2^{c\sqrt{\log |G| \log\log|G|}}$. This talk will outline the proof. Joint work with Matija Bucić, Tung Nguyen and Alex Scott.