On the Staircases of Gyárfás

In a 2011 paper, 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 $(4/5+\epsilon)n$ 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.