Erdős–Hajnal conjecture for H-free graphs

Prove that for every finite graph H, the class of H-free graphs has the Erdős–Hajnal property; that is, establish a constant e(H)>0 such that every H-free graph on n vertices contains a homogeneous vertex set of size at least n^{e(H)}.

Background

The paper introduces the Erdős–Hajnal property as the existence of polynomially large homogeneous vertex sets in every graph belonging to a class. It then identifies the Erdős–Hajnal conjecture as the assertion that this property holds for the class of H-free graphs for every fixed graph H.

The conjecture is presented as unresolved in its full generality, although the paper notes that it has recently been proved for all graphs H on five vertices. This is a general background conjecture rather than the central problem addressed by the paper, which instead proves a polynomial Ramsey statement for bipartite graphs of bounded VC-dimension.

References

The celebrated Erdős-Hajnal conjecture states that for each graph $H$ the class of $H$-free graphs admits the Erdős-Hajnal property . The conjecture is widely open in full, and was only recently resolved for all graphs on $5$ vertices .

— A Polynomial Ramsey Statement for Bounded VC-dimension  (2502.20461 - Hons, 27 Feb 2025) in Section 1, Introduction

Polynomial $\chi$-boundedness is particularly important because, if the class of $H$-free graphs is polynomially $\chi$-bounded, then $H$ satisfies the Erd\H{o}s--Hajnal conjecture~\, which asserts that there exists $\epsilon_H>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|V(G)|{\epsilon_H}$.

— Perfect Divisibility, Linear Divisibility and Chair-Free Graphs  (2608.14519 - Wang et al., 14 Aug 2026) in Section 1, Introduction