Extremal bounds for pattern avoidance in multidimensional 0-1 matrices

Abstract

A 0-1 matrix $M$ contains another 0-1 matrix $P$ if some submatrix of $M$ can be turned into $P$ by changing any number of $1$-entries to $0$-entries. $M$ is $\mathcal{P}$-saturated where $\mathcal{P}$ is a family of 0-1 matrices if $M$ avoids every element of $\mathcal{P}$ and changing any $0$-entry of $M$ to a $1$-entry introduces a copy of some element of $\mathcal{P}$. The extremal function $\operatorname{ex}(n,\mathcal{P})$ and saturation function $\operatorname{sat}(n,\mathcal{P})$ are the maximum and minimum possible weight of an $n\times n$ $\mathcal{P}$-saturated 0-1 matrix, respectively, and the semisaturation function $\operatorname{ssat}(n,P)$ is the minimum possible weight of an $n\times n$ $\mathcal{P}$-semisaturated 0-1 matrix $M$, i.e., changing any $0$-entry in $M$ to a $1$-entry introduces a new copy of some element of $\mathcal{P}$. We give upper bounds on parameters of minimally non-$O(n^{d-1})$ $d$-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-$O(n^{d-1})$ $d$-dimensional 0-1 matrices with all dimensions of length greater than $1$. For any positive integers $k,d$ and integer $r\in[0,d-1]$, we construct a family of $d$-dimensional 0-1 matrices with both extremal function and saturation function exactly $kn^r$ for sufficiently large $n$. We show that no family of $d$-dimensional 0-1 matrices has saturation function strictly between $O(1)$ and $\Theta(n)$ and we construct a family of $d$-dimensional 0-1 matrices with bounded saturation function and extremal function $\Omega(n^{d-\epsilon})$ for any $\epsilon>0$. Up to a constant multiplicative factor, we fully settle the problem of characterizing the semisaturation function of families of $d$-dimensional 0-1 matrices, which we prove to always be $\Theta(n^r)$ for some integer $r\in[0,d-1]$.