Galaxies and the Strong Erdos-Hajnal Property

TL;DR

This paper investigates the strong Erdős-Hajnal property in tournaments, proving it holds for certain structured tournaments where backedges form a forest of small trees, advancing understanding of the conjecture.

Contribution

It establishes the strong EH-property for a new class of tournaments characterized by their backedge structure forming a forest of small trees.

Findings

Proved the strong EH-property for tournaments with backedges forming a forest of small trees.

Extended the class of tournaments known to have the strong EH-property.

Connected structural properties of tournaments to the Erdős-Hajnal conjecture.

Abstract

An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon, Pack, and Solymosi states that for every tournament H there exists epsilon(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least |T|n and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-vertex tournament denoted by C5, where every vertex has two inneighbors and two outneighbors has the strong EH-property. It is known that every tournament with the strong EH-property also has the EH-property. In this paper we prove that tournaments that can be ordered in a way that the graph formed by the backedges is a forest consisting of trees with at most two edges and consecutive leaves under the vertex ordering has the strong EH-property.