Large unavoidable subtournaments

Let D_k denote the tournament on 3k vertices consisting of three disjoint vertex classes V₁, V₂ and V₃ of size k, each 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 D_k as a subtournament. Their proof showed that   and they conjectured that this could be reduced to n₀(k, ε) ⩽ ε^−O(k). Here we prove this conjecture.