A tournament approach to pattern avoiding matrices

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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(Tₙ,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: (1) Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n × n matrix with n(n)ᵒ⁽¹⁾ entries equal to 1 contains the pattern M. We show that this conjecture is equivalent to the assertion that t(Tₙ,H)=n(n)ᵒ⁽¹⁾ if and only if H belongs to a certain (natural) family of tournaments. (2) We propose an approach for determining if t(n,H)=n(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.