# Large homogeneous submatrices

## TL;DR

This paper proves that large zero-one matrices avoiding a specific small pattern necessarily contain large homogeneous submatrices, and characterizes which patterns guarantee submatrices of size nearly linear in the original matrix.

## Contribution

It provides a new structural result linking pattern avoidance in matrices to the existence of large homogeneous submatrices, with a near-complete classification of such patterns.

## Key findings

- Matrices avoiding certain patterns contain large homogeneous submatrices.
- Characterization of patterns guaranteeing near-linear size homogeneous submatrices.
- Applications to chordal bipartite graphs and other combinatorial structures.

## Abstract

A matrix is homogeneous if all of its entries are equal. Let $P$ be a $2\times 2$ zero-one matrix that is not homogeneous. We prove that if an $n\times n$ zero-one matrix $A$ does not contain $P$ as a submatrix, then $A$ has an $cn\times cn$ homogeneous submatrix for a suitable constant $c>0$. We further provide an almost complete characterization of the matrices $P$ (missing only finitely many cases) such that forbidding $P$ in $A$ guarantees an $n^{1-o(1)}\times n^{1-o(1)}$ homogeneous submatrix. We apply our results to chordal bipartite graphs, totally balanced matrices, halfplane-arrangements and string graphs.