The Removal Lemma for Tournaments

Suppose one needs to change the direction of at least $\epsilon n^2$ 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 $f^*_H(\epsilon)n^h$ 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 $f^*_H(\epsilon)$ is polynomial in $\epsilon$, 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-Szemer\'edi graphs. We finally show that even when restricted to tournaments, deciding if $H$ satisfies the condition of our characterization is an NP-hard problem.