Excluding pairs of tournaments
Krzysztof Choromański · Journal of Graph Theory · 2018
Abstract The Erdős–Hajnal conjecture states that for every given undirected graph H there exists a constant such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size at least . The conjecture is still open. Its equivalent directed version states that for every given tournament H there exists a constant such that every H‐free tournament T contains a transitive subtournament of order at least . In this article, we prove that for several pairs of tournaments, H1 and H2, there exists a constant such that every ‐free tournament T contains a transitive subtournament of size at least . In particular, we prove that for several tournaments H, there exists a constant such that every ‐free tournament T, where stands for the complement of H, has a transitive subtournament of size at least . To the best of our knowledge these are first nontrivial results of this type.