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arXiv · 1511.03504

On the staircases of Gyárfás

János Csányi·Peter Hajnal·Gábor V. Nagy
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Abstract

Gyárfás investigated a geometric Ramsey problem on convex, separated, balanced, geometric $K_{n,n}$. This led to appealing extremal problem on square $0$-$1$ matrices. Gyárfás 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 Gyárfás' 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 $(4/5+ε)n$ lower bound of Cai, Gyárfás 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.

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János Csányi, Peter Hajnal, Gábor V. Nagy. 2015-11-11. On the staircases of Gyárfás. https://arxiv.org/abs/1511.03504

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