Color critical hypergraphs and forbidden configurations

The present paper connects sharpenings of Sauer’s bound on forbidden configurations with color critical hypergraphs. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let F be a k × l (0,1)-matrix (the forbidden configuration). Assume A is an m × n simple matrix which has no submatrix which is a row and column permutation of F . We define forb(m, F ) as the best possible upper bound on n, for such a matrix A, which depends on m and F . It is known that forb(m, F ) = O(m) for any F , and Sauer’s bond states that forb(m, F ) = O(mk−1) fore simple F . We give sufficient condition for non-simple F to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.