The removal lemma for tournaments

Jacob Fox, Lior Gishboliner, Asaf Shapira, Raphael Yuster

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)110-134
Number of pages25
JournalJournal of Combinatorial Theory. Series B
StatePublished - May 2019

Bibliographical note

Funding Information:
Supported in part by ISF Grant 1028/16.

Funding Information:
Supported by a Packard Fellowship, by NSF CAREER award DMS 1352121, and by an Alfred P. Sloan Fellowship.Supported in part by ERC-Starting Grant 633509.Supported in part by ISF Grant 1028/16 and ERC-Starting Grant 633509.Supported in part by ISF Grant 1028/16.

Publisher Copyright:
© 2018 Elsevier Inc.


  • Removal lemma
  • Tournament

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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