Linear bounds on matrix extremal functions using visibility hypergraphs

TL;DR

This paper introduces a method using visibility hypergraphs to establish linear bounds on the extremal functions of 0-1 matrices avoiding certain submatrices, extending previous approaches with hypergraph techniques.

Contribution

It extends existing visibility graph methods to hypergraphs, providing new linear bounds on extremal functions for forbidden 0-1 matrices.

Findings

Established linear bounds on extremal functions using visibility hypergraphs

Extended previous bar visibility graph methods to hypergraphs

Provided bounds applicable to a broader class of forbidden matrices

Abstract

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 x n 0-1 matrix that avoids M, and let ex_k(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.