Exact Bounds for Forbidden Configurations and the Extremal Matrices

TL;DR

This paper establishes exact bounds on the maximum size of simple matrices avoiding certain configurations, specifically for a family of matrices with a particular pattern, and characterizes the extremal matrices for these bounds.

Contribution

It determines the exact maximum column counts for avoiding a specific configuration matrix for 1 ≤ p ≤ 9 and characterizes the extremal matrices, providing insights into forbidden configuration problems.

Findings

Exact bounds for forb(m, F(0,p,1,0)) for 1 ≤ p ≤ 9

Characterization of extremal matrices achieving these bounds

Potential extremal construction valid for all p

Abstract

Let $F$ be a $k\times \ell$ (0,1)-matrix. A matrix is simple if it is a (0,1)-matrix with no repeated columns. A (0,1)-matrix $A$ is said to have a $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace. Let $\mathrm{Avoid}(m,F)$ be all simple $m$-rowed matrices $A$ with no configuration $F$. Define $\mathrm{forb}(m,F)$ as the maximum number of columns of any matrix in $\mathrm{Avoid}(m,F)$. The $2\times (p+1)$ (0,1)-matrix $F(0,p,1,0)$ consists of a row of $p$ 1's and a row of one 1 in the remaining column. The paper determines $\mathrm{forb}(m,F(0,p,1,0))$ for $1\le p\le 9$ and the extremal matrices are characterized. A construction may be extremal for all $p$.