Large unavoidable subtournaments

© 2015 Elsevier B.V. Let Dk denote the tournament on 3k vertices consisting of three disjoint vertex classes V1, V2 and V3 of size k, each of which is oriented as a transitive subtournament, and with edges directed from V1 to V2, from V2 to V3 and from V3 to V1. Fox and Sudakov proved that given a natural number k and ε>0 there is n0(k, ε) such that every tournament of order n≥n0(k, ε) which is ε-far from being transitive contains Dk as a subtournament. Their proof showed that n0(k,ε)≤ε-O(k/ε2) and they conjectured that this could be reduced to n0(k, ε)≤ε-O(k). Here we outline a proof of this conjecture.