On the staircases of Gy\'arf\'as

TL;DR

This paper explores a non-symmetric variant of Gyárfás's geometric Ramsey problem, providing new bounds on staircase sizes in 0-1 matrices and resolving related extremal questions.

Contribution

It introduces the non-symmetric version of Gyárfás's problem and improves bounds on staircase sizes, including settling the problem for the sum of longest 0- and 1-staircases.

Findings

Established upper bounds for the extremal function.

Improved the lower bound from (4/5+ε)n to 5n/6-7/12.

Resolved the problem for the sum of the longest 0- and 1-staircases.

Abstract

Gy\'arf\'as investigated a geometric Ramsey problem on convex, separated, balanced, geometric $K_{n,n}$. This led to appealing extremal problem on square $0$-$1$ matrices. Gy\'arf\'as conjectured that any $0$-$1$ matrix of size $n\times n$ has a staircase of size $n-1$. We introduce the non-symmetric version of Gy\'arf\'as' problem. We give upper bounds and in certain range matching lower bound on the corresponding extremal function. In the square/balanced case we improve the $(4/5+\epsilon)n$ lower bound of Cai, Gy\'arf\'as et al. to $5n/6-7/12$. We settle the problem when instead of considering maximum staircases we deal with the sum of the size of the longest $0$- and $1$-staircases.