On the staircases of Gy\'arf\'as
J\'anos Cs\'anyi, Peter Hajnal, G\'abor V. Nagy

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 . This led to appealing extremal problem on square - matrices. Gy\'arf\'as conjectured that any - matrix of size has a staircase of size . 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 lower bound of Cai, Gy\'arf\'as et al. to . We settle the problem when instead of considering maximum staircases we deal with the sum of the size of the longest - and -staircases.
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