Pure pairs. VII. Homogeneous submatrices in 0/1-matrices with a forbidden submatrix

TL;DR

This paper investigates the size of homogeneous submatrices in 0/1-matrices avoiding a fixed submatrix, establishing near-linear bounds for acyclic forbidden matrices and advancing understanding of their structural properties.

Contribution

It proves that for any acyclic forbidden submatrix, the largest homogeneous submatrix size is almost linear in the matrix dimension, confirming a conjecture for a broad class of matrices.

Findings

Homogeneous submatrix size is n^{1-o(1)} for acyclic forbidden matrices.

Existence of large submatrices with dimensions Omega(n) by n^{1-o(1)} or vice versa.

Progress towards proving linear bounds for all acyclic forbidden matrices.

Abstract

For integer $n>0$, let $f(n)$ be the number of rows of the largest all-0 or all-1 square submatrix of $M$, minimized over all $n\times n$ $0/1$-matrices $M$. Thus $f(n)= O(\log n)$. But let us fix a matrix $H$, and define $f_H(n)$ to be the same, minimized over over all $n\times n$ $0/1$-matrices $M$ such that neither $M$ nor its complement (that is, change all $0$'s to $1$'s and vice versa) contains $H$ as a submatrix. It is known that $f_H(n)\ge \epsilon n^c$, where $c, \epsilon>0$ are constants depending on $H$. When can we take $c=1$? If so, then one of $H$ and its complement must be an acyclic matrix (that is, the corresponding bipartite graph is a forest). Korandi, Pach, and Tomon conjectured the converse, that $f_H(n)$ is linear in $n$ for every acyclic matrix $H$; and they proved it for certain matrices $H$ with only two rows. Their conjecture remains open, but we show $f_H(n)=n^{1-o(1)}$ for every acyclic matrix $H$; and indeed there is a $0/1$-submatrix that is either $\Omega(n)\times n^{1-o(1)}$ or $n^{1-o(1)}\times \Omega(n)$.