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 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
prove this conjecture.Comment: 9 page