Naturally submodular digraphs and forbidden digraph configurations

Small Forbidden Configurations III

The present paper continues the work begun by Anstee, Ferguson, Griggs and Sali on small forbidden configurations. 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 largest n, ...

Forbidden families of configurations

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 configuration if there is a submatrix of A which is a row and column permutation of F (trace is the set system version of a configuration). Let ‖A‖ denote the number of columns of A. Let F be a family of matrices. We define the extremal function forb(m,F) = max{‖A‖ : A is...

Small Forbidden Configurations II

The present paper continues the work begun by Anstee, Griggs and Sali on small forbidden configurations. In the notation of (0,1)-matrices, we consider a (0,1)-matrix F (the forbidden configuration), an m × n (0,1)-matrix A with no repeated columns which has no submatrix which is a row and column permutation of F , and seek bounds on n in terms of m and F . We give new exact bounds for some 2× ...

Linear Algebra and Forbidden Configurations

Pairwise intersections and forbidden configurations

By symmetry we can assume a ≥ d and b ≥ c. We show that fm(a, b, c, d) isΘ(ma+b−1) if either b > c or a, b ≥ 1. We also show that fm(0, b, b, 0) is Θ(m) and fm(a, 0, 0, d) is Θ(m). This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlsw...