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Published online by Cambridge University Press: 16 October 2017
This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For every k ≥ 1, every order-nk-dimensional permutation contains a monotone subsequence of length Ωk ( $\sqrt{n}$ ), 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 Θk (n k/(k+1)).