Extremal Functions of Forbidden Multidimensional Matrices

TL;DR

This paper advances the extremal theory of forbidden multidimensional matrices by establishing tight bounds on the maximum number of nonzero entries avoiding certain patterns, using combinatorics, probability, and analysis.

Contribution

It provides new bounds on extremal functions for multidimensional matrices, including tight bounds for tuple permutation matrices and super-homogeneity results.

Findings

Established $ heta(n^{d-1})$ bounds for block permutation matrices.

Proved super-homogeneity of extremal functions for certain matrices.

Improved bounds on the limit superior for permutation matrices.

Abstract

Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. 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 zeros. A fundamental problem is to study the maximum number of nonzero entries in a $d$-dimensional $n \times \cdots \times n$ matrix that avoids $P$. This maximum number, denoted by $f(n,P,d)$, is called the extremal function. We advance the extremal theory of matrices in two directions. The methods that we use come from combinatorics, probability, and analysis. Firstly, we obtain non-trivial lower and upper bounds on $f(n,P,d)$ when $n$ is large for every $d$-dimensional block permutation matrix $P$. We establish the tight bound $\Theta(n^{d-1})$ on $f(n,P,d)$ for every $d$-dimensional tuple permutation matrix $P$. This tight bound has the lowest possible order that an extremal function of a nontrivial matrix can ever achieve. Secondly, we show that $f(n,P,d)$ is super-homogeneous for a class of matrices $P$. We use this super-homogeneity to show that the limit inferior of the sequence $\{ {f(n,P,d) \over n^{d-1}}\}$ has a lower bound $2^{\Omega(k^{1/ d})}$ for a family of $k \times \cdots \times k$ permutation matrices $P$. We also improve the upper bound on the limit superior from $2^{O(k \log k)}$ to $2^{O(k)}$ for all $k \times \cdots \times k$ permutation matrices and show that the new upper bound also holds for tuple permutation matrices.