Forbidding Couples of Tournaments and the Erdös–Hajnal Conjecture

Abstract

A celebrated unresolved conjecture of Erdös 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 $n^{\epsilon (H)}$. 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 size at least $n^{\epsilon (H)}$. Recently the conjecture was proved for all six-vertex tournaments, except $K_{6}$. In this paper we construct two infinite families of tournaments for which the conjecture is still open for infinitely many tournaments in these two families—the family of so-called super nebulas and the family of so-called super triangular galaxies. We prove that for every super nebula $H_{1}$ and every $\varDelta$galaxy $H_{2}$ there exist $\epsilon (H_{1},H_{2})$ such that every $\lbrace H_{1},H_{2}\rbrace$-free tournament T contains a transitive subtournament of size at least $\mid T \mid ^{\epsilon (H_{1},H_{2})}$. We also prove that for every central triangular galaxy H there exist $\epsilon (K_{6},H)$ such that every $\lbrace K_{6},H\rbrace$-free tournament T contains a transitive subtournament of size at least $\mid T \mid ^{\epsilon (K_{6},H)}$. And we give an extension of our results.