  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 $0/1$ 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.
