On the Column Extremal Functions of Forbidden 0-1 Matrices

A 0-1 matrix is a matrix in which every element is either 0 or 1. The weight extremal function ex(n, P ) counts the maximum number of 1’s in an n × n matrix which avoids a pattern matrix P . The column extremal function exk(m,P ) counts the maximum number of columns that a matrix with m rows and k 1’s per column can contain such that the matrix avoids P . Set weight and column extremal functions count maximum numbers of 1’s and columns respectively of matrices which avoid a given collection of patterns. We find bounds on the column extremal function for elementary operations on one or two patterns. Using visibility representations, we determine linear bounds on the column extremal functions of patterns with 1’s in the same row crossing 1’s in the same column and linear bounds on the set extremal functions of a related class of pattern sets. We prove that for any r × c rectangular configuration, the column extremal function is θ(m). To improve and find new bounds on the weight extremal function, we determine the relation ex(m,n, P ) ≤ k(exk(m,P ) + n) for range-overlapping patterns P . We define a new pattern and use bounds on extremal functions of letter sequences coupled with matrix-sequence transformations to bound its column extremal function for k = 4 and 5. Finally, we find an upper bound on its weight extremal function by applying our inequality for range-overlapping patterns.