Extremal bounds for pattern avoidance in multidimensional 0-1 matrices

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. The 0-1 matrix M is $P$-saturated where $P$ is a family of 0-1 matrices if M avoids every element of $P$ and changing any 0-entry of M to a 1-entry introduces a copy of some element of $P$. The extremal function $\mathrm{ex}(n,P)$ and saturation function $\mathrm{sat}(n,P)$ are the maximum and minimum possible number of 1-entries in an $n\times n$ $P$-saturated 0-1 matrix, respectively, and the semisaturation function $\mathrm{ssat}(n,P)$ is the minimum possible number of 1-entries in an $n\times n$ $P$-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 $P$.

We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-$O\left(n^{d-1}\right)$ 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-$O\left(n^{d-1}\right)$ 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 $k{n}^r$ for sufficiently large n. We show that no family of d-dimensional 0-1 matrices has saturation function strictly between $O\left(1\right)$ and $\Theta \left(n\right)$ and we construct a family of d-dimensional 0-1 matrices with bounded saturation function and extremal function $\Omega \left(n^{d-ϵ}\right)$ for any $ϵ>0$. 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 $\Theta \left(n^r\right)$ for some integer $r\in [0,d-1]$.