## 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.

Original language | English |
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Pages (from-to) | 477-505 |

Number of pages | 29 |

Journal | Israel Journal of Mathematics |

Volume | 217 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2017 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)