EH-suprema of tournaments with no nontrivial homogeneous sets

Abstract

A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists ϵ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least nϵ(H).The conjecture has directed equivalent version stating that for every tournament H there exists ϵ(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least nϵ(H). For a fixed tournament H, define ξ(H) to be the supremum of all ϵ for which the following holds: for some n0 and every n>n0 every tournament with n≥n0 vertices not containing H as a subtournament has a transitive subtournament of size at least nϵ. We call ξ(H) the EH-supremum of H. The Erdös–Hajnal conjecture is true if and only if ξ(H)>0 for every H. If the conjecture is false then the smallest counterexample has no nontrivial so-called homogeneous sets (to be defined below). Therefore of interest are EH-suprema of those tournaments. In [5] it was proven that there exists a constant η>0 such that ξ(H)≤4h(1+ηlog⁡(h)h) for almost every h-vertex tournament H. However this result does not say anything about ξ(H) for an arbitrarily chosen tournament with no nontrivial homogeneous sets. We address that problem in this paper, proving that there exists C>0 such that every h-vertex tournament H with no nontrivial homogeneous sets satisfies ξ(H)≤Clog⁡(h)h. We will also give upper bounds on sizes of families of h-vertex tournaments with big EH-suprema. In [1] Alon, Pach and Solymosi proposed a procedure that produces bigger graphs satisfying the conjecture from smaller ones. All graphs obtained in such a way have nontrivial homogeneous sets. For a long time that was the only method to obtain infinite families of graphs satisfying the conjecture. Recently Berger, the author and Chudnovsky (see [2]) constructed a new infinite family of tournaments (so-called galaxies, to be defined below) that satisfies the conjecture and with no nontrivial homogeneous sets. Therefore it cannot be obtained by the procedure described in [1]. In this paper we construct a new infinite family of tournaments satisfying the conjecture – the family of so-called constellations (to be defined below). These results extend the results of [2] since every galaxy is a constellation.