Improved lower bounds on extremal functions of multidimensional permutation matrices

TL;DR

This paper extends the understanding of extremal functions of multidimensional permutation matrices, establishing new lower bounds for the maximum number of ones in large matrices avoiding a given permutation pattern.

Contribution

It generalizes previous two-dimensional results to higher dimensions, providing improved lower bounds for the extremal functions of permutation matrices in multiple dimensions.

Findings

For fixed dimension d ≥ 2, the maximum number of ones scales as 2^{k^{Θ(1)}} n^{d-1}.

Almost all d-dimensional permutation matrices exhibit this extremal behavior.

The results apply to sufficiently large matrices, broadening the scope of prior two-dimensional findings.

Abstract

A $d$-dimensional zero-one matrix $A$ avoids another $d$-dimensional zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeroes. Let $f(n,P,d)$ denote the maximum number of ones in a $d$-dimensional $n \times \cdots \times n$ zero-one matrix that avoids $P$. Fox proved for $n$ sufficiently large that $f(n, P, 2) = 2^{k^{\Theta(1)}}n$ for almost all $k \times k$ permutation matrices $P$. We extend this result by proving for $d \geq 2$ and $n$ sufficiently large that $f(n, P, d) = 2^{k^{\Theta(1)}}n^{d-1}$ for almost all $d$-dimensional permutation matrices $P$ of dimensions $k \times \cdots \times k$.