Towards the Erd\H{o}s-Hajnal conjecture for $P_5$-free graphs

The Erd\H{o}s-Hajnal conjecture is one of the most classical and well-known problems in extremal and structural combinatorics dating back to 1977. It asserts that in stark contrast to the case of a general $n$-vertex graph if one imposes even a little bit of structure on the graph, namely by forbidding a fixed graph $H$ as an induced subgraph, instead of only being able to find a polylogarithmic size clique or an independent set one can find one of polynomial size. Despite being the focus of considerable attention over the years the conjecture remains open. In this paper we improve the best known lower bound of $2^{\Omega(\sqrt{\log n})}$ on this question, due to Erd\H{o}s and Hajnal from 1989, in the smallest open case, namely when one forbids a $P_5$, the path on $5$ vertices. Namely, we show that any $P_5$-free $n$ vertex graph contains a clique or an independent set of size at least $2^{\Omega(\log n)^{2/3}}$. Our methods also lead to the same improvement for an infinite family of graphs.