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A tournament approach to pattern avoiding matrices

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Abstract

We consider the following Turán-type problem: given a fixed tournament H, what is the least integer t = t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t = t(T n ,H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several results on these problems, our main contributions are the following:

  • Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n × n matrix with n(log n)O(1) entries equal to 1 contains the pattern M. We show that this conjecture is equivalent to the assertion that t(T n ,H) = n(log n)O(1) if and only if H belongs to a certain (natural) family of tournaments.

  • We propose an approach for determining if t(n,H) = n(log n)O(1). This approach combines expansion in sparse graphs, together with certain structural characterizations of H-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach–Tardos conjecture.

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Authors and Affiliations

  1. School of Mathematics, Tel Aviv University, Tel Aviv, 69978, Israel

    Asaf Shapira

  2. Department of Mathematics, University of Haifa, Haifa, 31905, Israel

    Raphael Yuster

Authors
  1. Asaf Shapira
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  2. Raphael Yuster
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Corresponding author

Correspondence to Asaf Shapira.

Additional information

Supported in part by ISF Grant 224/11, Marie-Curie CIG Grant 303320 and ERC Starting Grant 633509.

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Shapira, A., Yuster, R. A tournament approach to pattern avoiding matrices. Isr. J. Math. 217, 477–505 (2017). https://doi.org/10.1007/s11856-017-1455-5

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  • DOI: https://doi.org/10.1007/s11856-017-1455-5

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