Induced subgraph density. V. All paths approach Erdos-Hajnal

TL;DR

This paper advances the understanding of the Erdős-Hajnal conjecture for paths, showing that all path-free graphs have large cliques or stable sets of super-polynomial size, nearly confirming the conjecture for all paths.

Contribution

It proves that for every path, $H$, $H$-free graphs have large homogeneous sets of size at least exponential in a sub-polynomial function of $n$, nearly confirming the Erdős-Hajnal conjecture for paths.

Findings

Confirmed the conjecture for paths with up to five vertices.

Established a near-optimal lower bound for clique or stable set size in $H$-free graphs.

Demonstrated that the size of these sets grows faster than any polynomial but slower than exponential in $n$.

Abstract

The Erd\H{o}s-Hajnal conjecture says that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a clique or stable set of size at least $n^c$. In this paper we are concerned with the case when $H$ is a path. The conjecture has been proved for paths with at most five vertices, but not for longer paths. We prove that the conjecture is ``nearly'' true for all paths: for every path $H$, all $H$-free graphs with $n$ vertices have cliques or stable sets of size at least $2^{(\log n)^{1-o(1)}}$.