Avoiding patterns in matrices via a small number of changes

TL;DR

This paper investigates the minimal number of entry modifications needed to eliminate a specific pattern in matrices with limited distinct entries, providing an asymptotically optimal solution for large matrices.

Contribution

It introduces a new function measuring pattern avoidance in matrices and derives an asymptotically tight value for this function in the case of matrices with limited entries.

Findings

Derived an asymptotically tight value for the pattern-avoidance function.

Established bounds for the minimum changes needed in matrices with limited entries.

Provided insights into the structure of matrices avoiding certain patterns.

Abstract

Let ${\cal A}=\{A_1,\ldots, A_r\}$ be a partition of a set $\{1,\ldots,m\}\times\{1,\ldots, n\}$ into $r$ nonempty subsets, and $A=(a_{ij})$ be an $m\times n$ matrix. We say that $A$ has a pattern ${\cal A}$ provided that $a_{ij}=a_{i'j'}$ if and only if $(i,j),(i',j')\in A_t$ for some $t\in\{1,\ldots,r\}$. In this note we study the following function $f$ defined on the set of all $m\times n$ matrices $M$ with $s$ distinct entries: $f(M; {\cal A})$ is the smallest number of positions where the entries of $M$ need to be changed such that the resulting matrix does not have any submatrix with pattern ${\cal A}$. We give an asymptotically tight value for $$ f(m,n; s, {\cal A}) = \max\{f(M; {\cal A}): M \mbox{ is an } m\times n\mbox{ matrix with at most } s \mbox{ distinct entries}\} . $$