Linear bounds on matrix extremal functions using visibility hypergraphs

The 0− 1 matrix A contains a 0− 1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeroes. If A does not contain M , then A avoids M . Let ex(n,M) be the maximum number of ones in an n×n 0−1 matrix that avoids M , and let exk(m,M) be the maximum number of columns in a 0− 1 matrix with m rows that avoids M and has at least k ones in every column. A method for bounding ex(n,M) by using bounds on the maximum number of edges in bar visibility graphs was introduced in (R. Fulek, Discrete Mathematics 309, 2009). By using a similar method with bar visibility hypergraphs, we obtain linear bounds on the extremal functions of other forbidden 0− 1 matrices.