Exponential Erdős–Szekeres theorem for matrices

Abstract In 1993, Fishburn and Graham established the following qualitative extension of the classical Erdős–Szekeres theorem. If is sufficiently large with respect to , then any real matrix contains an submatrix in which every row and every column is monotone. We prove that the smallest such is at most , greatly improving the previously best known double‐exponential upper bound of Bucić, Sudakov, and Tran, and matching the best known lower bound on an exponential scale. In particular, we prove the following surprisingly sharp transition in the asymmetric setting. On one hand, every matrix contains an submatrix, in which every row is monotone. On the other hand, there exist matrices containing no such submatrix.