Lexicographic Ramsey theory (Q1803874)

The following extension of the Erdős-Szekeres theorem is proved. If \(d,n\) are positive integers then there exists an integer \(N\) such that if \(f\) is an injection from \(\{1,2,\dots,N\}^ d\) into the reals then there is an \(n\times\cdots\times n\) subcube on which \(f\) is lexicographic and monotonic on each coordinate. The lower bound \(N>n^{(1-1/d)n^{d-1}}\) is given.