Multi-symbol forbidden configurations

Abstract

An r-matrix is a matrix with symbols in {0, 1, . . . , r − 1}. A matrix is simple if it has no repeated columns. Let the support of a matrix F , supp(F ) be the largest simple matrix such that every column in supp(F ) is in F . For a family of r-matrices F, we define forb(m, r, F) as the maximum number of columns of an m-rowed, r-matrix A such that F is not a row-column permutation of A for all F ∈ F. While many results exist for r = 2, there are fewer for larger numbers of symbols. We expand on the field of forbidding matrices with r-symbols, introducing a new construction for lower bounds of the growth of forb(m, r, F) (with respect to m) that is applicable to matrices that are either not simple or have a constant row. We also introduce a new upper bound restriction that helps with avoiding non-simple matrices, limited either by the asymptotic bounds of the support, or the size of the forbidden matrix, whichever is larger. Continuing the trend of upper bounds, we represent a well-known technique of standard induction as a graph, and use graph theory methods to obtain asymptotic upper bounds. With these techniques we solve multiple, previously unknown, asymptotic bounds for a variety of matrices. Finally, we end with block matrices, or matrices with only constant row, and give bounds for all possible cases.

Exponential multivalued forbidden configurations

“…On the other hand, the more general case of r-matrices is not so well-explored. Previous papers mainly focus on providing bounds on the forbidden number for special classes of sets in the polynomial case [4,5]. In this paper, we dive into exponential forbidden numbers and provide exact bounds when (0, 1)-configurations of r-matrices are forbidden.…”