An exact characterization of saturation for permutation matrices

TL;DR

This paper provides a complete classification of the saturation functions for permutation matrices, determining the exact growth behavior of the minimum number of 1s in matrices avoiding a given permutation pattern.

Contribution

It offers an exact characterization of saturation functions specifically for permutation matrices, resolving a classification problem in combinatorics.

Findings

Saturation functions for permutation matrices are fully classified.

Saturation functions are either constant or linear in n.

The paper establishes the precise conditions determining each case.

Abstract

A 0-1 matrix $M$ contains a 0-1 matrix pattern $P$ if we can obtain $P$ from $M$ by deleting rows and/or columns and turning arbitrary 1-entries into 0s. The saturation function $\mathrm{sat}(P,n)$ for a 0-1 matrix pattern $P$ indicates the minimum number of 1s in an $n \times n$ 0-1 matrix that does not contain $P$, but changing any 0-entry into a 1-entry creates an occurrence of $P$. Fulek and Keszegh recently showed that each pattern has a saturation function either in $O(1)$ or in $\Theta(n)$. We fully classify the saturation functions of permutation matrices.