Multi-symbol forbidden configurations

An r-matrix is a matrix with symbols in {0,1,…,r−1}. A matrix is simple if it has no repeated columns. For a family of r-matrices F, we define forb(m,r,F) as the maximum number of columns of an m-rowed, simple r-matrix A such that F is not a row–column permutation of a submatrix of A for all F∈F. We investigate the asymptotic bounds of special families of forbidden configurations Sym(F), a symmetric family of matrices based on a (0,1)-matrix F. The previously known lower bound constructions required F to have no constant row. We introduce a new lower bound construction that drops this requirement and is a better bound in certain cases. We will also investigate the effects of constant rows from the upper bound perspective, including upper bounds of block matrices, matrices consisting of entirely constant rows.