Saturation problems about forbidden $0$-$1$ submatrices

TL;DR

This paper investigates the minimal number of ones in large 0-1 matrices that are saturating for forbidden submatrices, establishing bounds and classifications for various matrix types and extending prior research in the field.

Contribution

It extends the study of saturation functions for 0-1 matrices, proving dichotomies, providing bounds for specific matrices, and classifying matrices with linear saturation functions.

Findings

Saturation function for forbidden matrices is either constant or linear in size.

Identified classes of matrices with linear saturation functions.

Constructed a 5x5 permutation matrix with bounded saturation function.

Abstract

A $0$-$1$ matrix $M$ is saturating for a $0$-$1$ matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by changing some $1$ entries to $0$ entries, and changing an arbitrary $0$ to $1$ in $M$ introduces such a submatrix in $M$. In saturation problems for $0$-$1$ matrices we are interested in estimating the minimum number of $1$ entries in an $m \times n$ matrix that is saturating for $P$, in terms of $m$ and $n$. In other words, we wish to give good estimates for the saturation function of $P$. Recently, Brualdi and Cao initiated the study of saturation problems in the context of $0$-$1$ matrices. We extend their work in several directions. We prove that every $0$-$1$ forbidden matrix has its saturation function either in $\Theta(1)$ or $\Theta(n)$ in the case when we restrict ourselves to square saturating matrices. Then we give a partial answer to a question posed by Brualdi and Cao about the saturation function of $J_k$, which is obtained from the identity matrix $I_k$ by putting the first row after the last row. Furthermore, we exhibit a $5\times 5$ permutation matrix with the saturation function bounded from the above by a fixed constant. We complement this result by identifying large classes of $0$-$1$ matrices with linear saturation function. Finally, we completely resolve the related semisaturation problem as far as the constant vs. linear dichotomy is concerned.