The Strong EH-Property and the Erd\H{o}s-Hajnal Conjecture

TL;DR

This paper introduces the strong EH-property, a new method for proving the Erd ext{"o}s-Hajnal} Conjecture for prime tournaments, and demonstrates how it can be used to construct larger tournaments satisfying the conjecture.

Contribution

The paper presents the strong EH-property, enabling the combination of tournaments satisfying the conjecture to form larger ones, including nonprime tournaments, advancing the understanding of the conjecture.

Findings

Established the strong EH-property for certain tournaments.

Provided examples of tournament families constructed via the new method.

Suggested potential for proving the conjecture for new classes of tournaments.

Abstract

The Erd\H{o}s-Hajnal Conjecture states that for every $H$ there exists a constant $\epsilon(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $|V(G)|^{\epsilon(H)}$. The Conjecture is still open. Some time ago its directed version was formulated (see:\cite{alon}). In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. If the Conjecture is not true then the smallest counterexample is a prime tournament. For a long time the Conjecture was known only for finitely many prime tournaments. Recently in \cite{bcc} and \cite{choromanski2} the Conjecture was proven for the families of galaxies and constellations that contain infinitely many prime tournaments. In \cite{bcc} the Conjecture was also proven for all $5$-vertex tournaments. We say that a tournament $H$ has the $EH$-property if it satisfies the Conjecture. In this paper we introduce the so-called \textit{strong EH-property} which enables us to prove the Conjecture for new prime tournaments, but what is even more interesting, provides a mechanism to combine tournaments satisfying the Conjecture to get bigger tournaments that do so and are not necessarily nonprime. We give several examples of families of tournaments constructed according to this procedure. The only procedure known before used to construct bigger tournaments satisfying the Conjecture from smaller tournaments satisfying the Conjecture was the so-called \textit{substitution procedure} (see: \cite{alon}). However an outcome of this procedure is always a nonprime tournament and, from what we have said before, prime tournaments are those that play crucial role in the research on the Conjecture. Our method may be potentially used to prove the Conjecture for several new classes of tournaments.