About the Erd\"{o}s-Hajnal conjecture for seven-vertex tournaments

Abstract

A celebrated unresolved conjecture of Erd\"{o}s and Hajnal 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 \( n^{\epsilon(H)} \). The 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 \( n^{\epsilon(H)} \). Both the directed and the undirected versions of the conjecture are known to be true for small graphs (tournaments). So far the conjecture was proved only for some specific families of prime tournaments, tournaments constructed according to the so\(-\)called substitution procedure allowing to build bigger graphs, and for all five\(-\)vertex tournaments. Recently the conjecture was proved for all six\(-\)vertex tournament, with one exception, but the question about the correctness of the conjecture for all seven\(-\)vertex tournaments remained open. In this paper we prove the correctness of the conjecture for several seven\(-\)vertex tournaments.