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 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 ǫ > 0 there is n0(k, ǫ) such that every tournament of order n ≥ n0(k, ǫ) which is ǫ-far from being transitive contains Dk as a subtournament. Their proof showed that n0(k, ǫ) ≤ ǫ −O(k/ǫ) and they conjectured that this could be reduced to n0(k, ǫ) ≤ ǫ . Here we prove this conjecture.