Leírás
An $r$-matrix is a matrix with symbols in $\{0,1,\dots,r-1\}$. 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 $\mathcal{F}$, we define $\forb(m,r,\mathcal{F})$ 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 $F \in \mathcal{F}$. 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 that is appicable to matrices that are either not simple or have a constant row. We also introduce a new upper bound for non-simple matrices, limited either by the asymptotic bounds of the support, or the size of the forbidden matrix, whichever is larger. We represent a well-known technique of standard induction as a graph, and use graph theory methods to obtain upper bound. We end with block matrices, or matrices with only constant rows, and give bounds for all possible cases.