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

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.