Monotone Subsequences in High-Dimensional Permutations

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdos-Szekeres theorem: For every k ≥ 1, every order-n k-dimensional permutation contains a monotone subsequence of length Ω k , and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θ k(n ^k/(k+1)).