The removal lemma for tournaments

Suppose one needs to change the direction of at least εn ² edges of an n-vertex tournament T, in order to make it H-free. A standard application of the regularity method shows that in this case T contains at least f _H ^⁎ (ε)n ^h copies of H, where f _H ^⁎ is some tower-type function. It has long been observed that many graph/digraph problems become easier when assuming that the host graph is a tournament. It is thus natural to ask if the removal lemma becomes easier if we assume that the digraph G is a tournament. Our main result here is a precise characterization of the tournaments H for which f _H ^⁎ (ε) is polynomial in ε stating that such a bound is attainable if and only if H's vertex set can be partitioned into two sets, each spanning an acyclic directed graph. The proof of this characterization relies, among other things, on a novel application of a regularity lemma for matrices due to Alon, Fischer and Newman, and on probabilistic variants of Ruzsa–Szemerédi graphs. We finally show that even when restricted to tournaments, deciding if H satisfies the condition of our characterization is an NP-hard problem.