The Erd\H{o}s-Hajnal Conjecture for Paths and Antipaths

TL;DR

This paper proves a significant case of the Erdős-Hajnal conjecture, showing that graphs excluding a path and its complement contain large homogeneous sets, advancing understanding of graph structure.

Contribution

It establishes the Erdős-Hajnal property for paths and antipaths, providing bounds on clique or stable set sizes in such graphs.

Findings

Graphs excluding a path and its complement have large homogeneous sets.

Existence of a positive constant c_k for each path length k.

Improves understanding of graph structure related to the Erdős-Hajnal conjecture.

Abstract

We prove that for every k, there exists $c_k>0$ such that every graph G on n vertices not inducing a path $P_k$ and its complement contains a clique or a stable set of size $n^{c_k}$.