Galaxies and the Strong Erdos-Hajnal Property
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.
Keywords
Cite
@article{arxiv.2207.02483,
title = {Galaxies and the Strong Erdos-Hajnal Property},
author = {Soukaina Zayat},
journal= {arXiv preprint arXiv:2207.02483},
year = {2023}
}
Comments
The result in this paper is a very special case of the result I proved in the article (2207.09146), entitled: "Forests and the Strong Erdos-Hajnal Property". When I submitted the generalized version I didn't know that I can just replace the old version with the new one