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

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.