Forbidden Configurations and Boundary Cases

TL;DR

This paper investigates the maximum size of simple (0,1)-matrices avoiding a specific configuration, providing exact bounds and constructions for certain boundary cases where the maximum size jumps significantly.

Contribution

It characterizes the exact maximum number of columns in simple matrices avoiding a configuration for boundary cases with a sharp size increase.

Findings

Determined exact bounds for certain boundary configurations.

Identified matrices with size bounds of Θ(m^{k-2}) and Ω(m^{k-1}).

Provided constructions achieving these bounds.

Abstract

Let $F$ be a $k\times \ell$ (0,1)-matrix. Define a (0,1)-matrix $A$ to have a $F$ as a \emph{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 \emph{trace}. Define a matrix to be {\it simple} if it is a (0,1)-matrix with no repeated columns. 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 $\mathrm{Avoid}(m,F)$. Determining $\mathrm{forb}(m,F)$ requires determining bounds and constructions of matrices in $\mathrm{Avoid}(m,F)$. The paper considers some column maximal $k$-rowed simple $F$ that have the bound $\Theta(m^{k-2})$ and yet adding a column increases bound to $\Omega(m^{k-1})$. By a construction, $\mathrm{forb(m,F)}$ is determined exactly.