Forests and the Strong Erdos-Hajnal Property

TL;DR

This paper introduces a new class of tournaments called spiral galaxies and proves they possess the strong Erdős-Hajnal property, advancing understanding of the structure of H-free tournaments.

Contribution

The paper constructs an infinite class of tournaments, spiral galaxies, and proves they have the strong EH-property, expanding the classes of tournaments known to have this property.

Findings

Spiral galaxies have the strong EH-property.

Every spiral galaxy contains large homogeneous subsets.

The result extends the class of tournaments with the strong EH-property.

Abstract

An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon et al. 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 c|T| 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 construct an infinite class of tournaments - the so-called spiral galaxies and we prove that every spiral galaxy has the strong EH-property.