A tournament approach to pattern avoiding matrices

TL;DR

This paper investigates the minimum number of edge additions needed to embed a fixed tournament into a larger tournament, linking this problem to a conjecture on pattern-avoiding matrices and proposing a structural approach for its resolution.

Contribution

It establishes an equivalence between a matrix pattern conjecture and a tournament edge addition problem, and introduces a structural method to determine the order of the minimal edge additions.

Findings

Proves the equivalence between Pach and Tardos conjecture and tournament edge addition thresholds.

Proposes a structural approach combining graph expansion and tournament characterization.

Opens new avenues for applying graph theory to matrix pattern avoidance conjectures.

Abstract

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.