Large unavoidable subtournaments

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.