Forbidden Submatrices: Some New Bounds and Constructions

We explore an extremal hypergraph problem for which both the vertices and edges are ordered. Given a hypergraph $F$ (not necessarily simple), we consider how many edges a simple hypergraph (no repeated edges) on $m$ vertices can have while forbidding $F$ as a subhypergraph where both hypergraphs have fixed vertex and edge orderings. A hypergraph of $n$ edges on $m$ vertices can be encoded as an $m\times n$ (0,1)-matrix. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given a (0,1)-matrix $F$, we define ${\hbox{fs}}(m,F)$ as the maximum, over all simple matrices $A$ which do not have $F$ as a submatrix, of the number of columns in $A$. The row and column order matter. It is known that if $F$ is $k\times \ell$ then ${\hbox{fs}}(m,F)$ is $O(m^{2k-1-\epsilon})$ where $\epsilon=(k-1)/(13\log_2 \ell)$. Anstee, Frankl, Füredi and Pach have conjectured that if $F$ is $k$-rowed, then ${\hbox{fs}}(m,F)$ is $O(m^k)$. We show ${\hbox{fs}}(m,F)$ is $O(m^2)$ for $F= \left[{1\,0\,1\,0\,1\atop 0\,1\,0\,1\,0}\cdots\right]$ and for $F= \left[{1\,0\,1\,0\,1\atop 1\,0\,1\,0\,1}\cdots\right]$. The proofs use a type of amortized analysis. We also give some constructions.