The Erd\H{o}s-Hajnal Conjecture for Long Holes and Anti-holes

TL;DR

This paper proves a special case of the Erd ext{"o}s-Hajnal conjecture, showing that graphs avoiding long cycles and their complements have large homogeneous sets, advancing understanding of graph structure related to the conjecture.

Contribution

It establishes the Erd ext{"o}s-Hajnal property for graphs excluding long holes and anti-holes, a significant step in the conjecture's broader context.

Findings

Graphs excluding long cycles and their complements have large cliques or stable sets.

Existence of a positive constant $c_k$ for such graphs ensuring large homogeneous sets.

Progress towards the Erd ext{"o}s-Hajnal conjecture for specific graph classes.

Abstract

Erd\H{o}s and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$, there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size $n^{c_k}$.