A tournament approach to pattern avoiding matrices

We consider the following Tur'an-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: (1) Pach and Tardos conjectured that if $M$ is an acyclic $0\mathrm{/}1$ matrix, then any $n\times n$ 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. (2) 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.