Forbidden Berge Hypergraphs

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F , we say that a (0,1)-matrix A has F as a Berge hypergraph if there is a submatrix B of A and some row and column permutation of F , say G, with G 6 B. Letting ‖A‖ denote the number of columns in A, we define the extremal function Bh(m,F ) = max{‖A‖ : A m-rowed simple matrix and no Berge hypergraph F}. We determine the asymptotics of Bh(m,F ) for all 3and 4-rowed F and most 5-rowed F . For certain F , this becomes the problem of determining the maximum number of copies of Kr in a m-vertex graph that has no Ks,t subgraph, a problem studied by Alon and Shikhelman.