Exact Bounds for Forbidden Configurations and the Extremal Matrices

Let $F$ be a $k\times \ell$ (0,1)-matrix. A matrix is simple if it is a (0,1)-matrix with no repeated columns. A (0,1)-matrix $A$ is said to have a $F$ as a 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 trace. 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 $\mathrm{Avoid}(m,F)$. The $2\times (p+1)$ (0,1)-matrix $F(0,p,1,0)$ consists of a row of $p$ 1's and a row of one 1 in the remaining column. The paper determines $\mathrm{forb}(m,F(0,p,1,0))$ for $1\le p\le 9$ and the extremal matrices are characterized. A construction may be extremal for all $p$.