Almost all permutation matrices have bounded saturation functions

TL;DR

This paper proves that almost all permutation matrices have a bounded minimum number of ones in saturating matrices, advancing understanding of saturation functions for forbidden submatrices in combinatorics.

Contribution

It confirms that almost all permutation matrices have bounded saturation functions, and provides a family of such matrices, addressing a conjecture and progress on characterization.

Findings

Almost all permutation matrices have bounded saturation functions.

The paper exhibits a family of permutation matrices with bounded saturation functions.

Progress on the characterization of matrices with bounded saturation functions.

Abstract

Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that 0-1 matrix $A$ is saturating for the forbidden 0-1 matrix $P$ if $A$ avoids $P$ but changing any zero to a one in $A$ creates a copy of $P$. Define $sat(n, P)$ to be the minimum possible number of ones in an $n \times n$ 0-1 matrix that is saturating for $P$. Fulek and Keszegh proved that for every 0-1 matrix $P$, either $sat(n, P) = O(1)$ or $sat(n, P) = \Theta(n)$. They found two 0-1 matrices $P$ for which $sat(n, P) = O(1)$, as well as infinite families of 0-1 matrices $P$ for which $sat(n, P) = \Theta(n)$. Their results imply that $sat(n, P) = \Theta(n)$ for almost all $k \times k$ 0-1 matrices $P$. Fulek and Keszegh conjectured that there are many more 0-1 matrices $P$ such that $sat(n, P) = O(1)$ besides the ones they found, and they asked for a characterization of all permutation matrices $P$ such that $sat(n, P) = O(1)$. We affirm their conjecture by proving that almost all $k \times k$ permutation matrices $P$ have $sat(n, P) = O(1)$. We also make progress on the characterization problem, since our proof of the main result exhibits a family of permutation matrices with bounded saturation functions.