Forbidden Families of Minimal Quadratic and Cubic Configurations

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 $\mathrm{\mid }A\mathrm{\mid }$ 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}$(m, F_i)$hasgreaterasymptoticgrowththan$$\text{forb}$(m, $\mathcal{F}$)$, extending research started by Anstee and Koch.