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

Erdős and Hajnal posed the following conjecture at the intersection of graph Ramsey theory and structural graph theory. \begin{conjecture}[Erdős--Hajnal]\label[conjecture]{conj:EH} For every graph $H$, the class of $H$-free graphs has the Erdős--Hajnal property. \end{conjecture}

The conjecture predicts a local-to-global phenomenon: excluding one fixed induced subgraph forces a polynomial-size clique or independent set. It remains wide open and appears difficult even for small graphs $H$.

— Induced packing treewidth II. Excluding a clique or a biclique  (2609.21615 - Nikabadi et al., 18 Sep 2026) in Section 1, paragraph “Erdős–Hajnal property,” Conjecture “Erdős–Hajnal” (Conjecture \ref{conj:EH})

Of the eight remaining pairs, two have no member containing an induced P5 or P5, while six do. To the best of our knowledge, all eight remain open as of 26. August 2026.

— The Erdős-Hajnal Property for the six-vertex Graph with Edge Set $\{ab,bc,cd,de,af,bf,df\}$  (2608.28551 - Tran et al., 28 Aug 2026) in Section 1.2, p. 3