  In 1993, Fishburn and Graham established the following qualitative extension
of the classical Erd\H{o}s-Szekeres theorem. If $N$ is sufficiently large with
respect to $n$, then any $N\times N$ real matrix contains an $n\times n$
submatrix in which every row and every column is monotone. We prove that the
smallest such $N$ is at most $2^{n^{4+o(1)}}$, greatly improving the previously
best known double-exponential upper bound, and getting close to the best known
lower bound $n^{n/2}$.
  In particular, we prove the following surprising sharp transition in the
asymmetric setting. On one hand, every $8n^2\times 2^{n^{4+o(1)}}$ matrix
contains an $n\times n$ submatrix, in which every row is mononote. On the other
hand, there exist $n^{2}/6\times 2^{2^{n^{1-o(1)}}}$ matrices containing no
such submatrix .
