Excluding pairs of tournaments

Abstract

The Erdős–Hajnal conjecture states that for every given undirected graph H there exists a constant c(H)>0 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 false|V(G)false|cfalse(Hfalse). The conjecture is still open. Its equivalent directed version states that for every given tournament H there exists a constant c(H)>0 such that every H‐free tournament T contains a transitive subtournament of order at least false|V(T)false|cfalse(Hfalse). In this article, we prove that for several pairs of tournaments, H1 and H2, there exists a constant c(H1,H2)>0 such that every {H1,H2}‐free tournament T contains a transitive subtournament of size at least false|V(T)false|cfalse(H1,H2false). In particular, we prove that for several tournaments H, there exists a constant c(H)>0 such that every {H,Hc}‐free tournament T, where Hc stands for the complement of H, has a transitive subtournament of size at least false|V(T)false|cfalse(Hfalse). To the best of our knowledge these are first nontrivial results of this type.

Excluding Hooks and their Complements

“…The question of excluding pairs of graphs in the context of the Erdős-Hajnal conjecture was considered also in the directed setting (see: [9]). The directed version of the conjecture is equivalent to the undirected one and was recently heavily investigated ( [3,11,13,10,12]).…”