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The Removal Lemma for Tournaments
Suppose one needs to change the direction of at least edges of
an -vertex tournament , in order to make it -free. A standard
application of the regularity method shows that in this case contains at
least copies of , where 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 is a
tournament.
Our main result here is a precise characterization of the tournaments for
which is polynomial in , stating that such a bound
is attainable if and only if '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
satisfies the condition of our characterization is an NP-hard problem