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 $Fprec A$, 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 $text{forb}(m, mathcal{F}) = max{|A|colon A text{ is an }m-text{rowed simple matrix and has no configuration } Finmathcal{F}}$. 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.