Exponential multivalued forbidden configurations

TL;DR

This paper investigates the maximum size of $(0,1)$-matrices avoiding certain submatrix patterns, extending the concept to matrices with entries in $oxed{0,1, ext{...},r-1}$, and provides bounds and stability results for these generalized forbidden configurations.

Contribution

It offers exact bounds for forbidden numbers in generalized matrices and reveals qualitative differences across different entry sets, especially for $r > 3$.

Findings

Exact bounds for many $(0,1)$-matrices $F$.

Stability result for the $2\times 2$ identity matrix.

Differences between cases $r=2$, $r=3$, and $r>3$.

Abstract

The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.