Forbidden Families of Minimal Quadratic and Cubic Configurations

TL;DR

This paper investigates forbidden configurations in simple (0,1)-matrices, establishing that pairs of configurations with no shared extremal structures lead to smaller combined extremal functions than individual ones, extending previous combinatorial research.

Contribution

It characterizes pairs of forbidden configurations with distinct extremal structures and shows their combined extremal function grows slower than each individually.

Findings

Pairs with no common extremal construction reduce the extremal function.

Each configuration's extremal function grows faster than the combined case.

Extends previous work on forbidden configurations in combinatorics.

Abstract

A matrix is \emph{simple} if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a \emph{configuration}, denoted $F\prec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $\mathcal{F}$ be a family of matrices. We define the extremal function $\text{forb}(m, \mathcal{F}) = \max\{|A|\colon A \text{ is an }m-\text{rowed simple matrix and has no configuration } F\in\mathcal{F}\}$. We consider pairs $\mathcal{F}=\{F_1,F_2\}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $\text{forb}(m, F_i)$ has greater asymptotic growth than $\text{forb}(m, \mathcal{F})$, extending research started by Anstee and Koch.