  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.
