Saturation of multidimensional 0-1 matrices

TL;DR

This paper extends the study of saturation and semisaturation functions from two-dimensional to multidimensional 0-1 matrices, providing exact values for certain cases and conditions for boundedness.

Contribution

It generalizes known results to multidimensional matrices, finds exact saturation and extremal values for identity matrices, and characterizes when semisaturation functions are bounded.

Findings

Exact values of ex(n;P,d) and sat(n;P,d) for multidimensional identity matrices.

Necessary and sufficient conditions for bounded semisaturation functions in multidimensional matrices.

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 flipping any number of its $1$-entries to $0$-entries, and changing any $0$-entry to $1$-entry of $M$ introduces a copy of $P$. Matrix $M$ is semisaturating for $P$ if changing any $0$-entry to $1$-entry of $M$ introduces a new copy of $P$, regardless of whether $M$ originally contains $P$ or not. The functions $ex(n;P)$ and $sat(n;P)$ are the maximum and minimum possible number of $1$-entries a $n\times n$ 0-1 matrix saturating for $P$ can have, respectively. Function $ssat(n;P)$ is the minimum possible number of $1$-entries a $n\times n$ 0-1 matrix semisaturating for $P$ can have. Function $ex(n;P)$ has been studied for decades, while investigation on $sat(n;P)$ and $ssat(n;P)$ was initiated recently. In this paper, we make nontrivial generalization of results regarding these functions to multidimensional 0-1 matrices. In particular, we find the exact values of $ex(n;P,d)$ and $sat(n;P,d)$ when $P$ is a $d$-dimensional identity matrix. Then we give the necessary and sufficient condition for a multidimensional 0-1 matrix to have bounded semisaturation function.