Forbidden Families of Configurations

TL;DR

This paper investigates the maximum number of columns in simple (0,1)-matrices with a fixed number of rows that avoid certain configurations, focusing on families where individual constraints grow faster than combined ones.

Contribution

It introduces the concept of forbidden configuration families and analyzes their extremal functions, revealing cases where combined restrictions result in slower growth than individual ones.

Findings

Identifies families of configurations with unique extremal growth behaviors.

Shows that combined configuration constraints can lead to slower growth rates.

Provides new bounds for extremal functions in matrix configuration avoidance.

Abstract

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$ (trace is the set system version of a configuration). Let $\ncols{A}$ denote the number of columns of $A$. Let ${\cal F}$ be a family of matrices. We define the extremal function $forb(m,{\cal F})=\max{\ncols{A} : A is m-rowed simple matrix and has no configuration F\in{\cal F}}$. We consider some families ${\cal F}=\{F_1,F_2,\ldots, F_t\}$ such that individually each $\forb(m,F_i)$ has greater asymptotic growth than $\forb(m,{\cal F})$.