Erd\"{o}s-Hajnal Conjecture for New Infinite Families of Tournaments

TL;DR

This paper proves the Erdős-Hajnal conjecture for two new infinite families of tournaments, called galaxies with spiders and asterisms, expanding the classes of tournaments for which the conjecture holds.

Contribution

The paper introduces two new infinite families of tournaments and establishes the Erdős-Hajnal conjecture for these classes, advancing understanding of the conjecture's scope.

Findings

Proved the conjecture for galaxies with spiders.

Proved the conjecture for asterisms.

Expanded classes of tournaments satisfying the conjecture.

Abstract

Erd\"{o}s-Hajnal conjecture states that for every undirected graph $H$ there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{\epsilon(H)} $. This conjecture has a directed equivalent version stating 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)} $. This conjecture is known to hold for a few infinite families of tournaments. In this paper we construct two new infinite families of tournaments - the family of so-called galaxies with spiders and the family of so-called asterisms, and we prove the correctness of the conjecture for these two families.