Tournaments and the Erd\"{o}s-Hajnal Conjecture

TL;DR

This paper proves the Erdős-Hajnal conjecture for a new infinite family of tournaments called flotilla-galaxies, advancing understanding of the conjecture's validity in directed graphs.

Contribution

The paper introduces flotilla-galaxies, a new infinite family of tournaments, and proves the Erdős-Hajnal conjecture for all of them.

Findings

Proved the conjecture for flotilla-galaxies.

Constructed a new family of tournaments.

Extended the class of tournaments satisfying the conjecture.

Abstract

The celebrated 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 proved for few infinite families of tournaments. In this paper we construct a new infinite family of tournaments $-$ the family of so-called flotilla-galaxies and we prove the correctness of the conjecture for every flotilla-galaxy tournament.