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Monotone Subsequences in High-Dimensional Permutations

Part of: Designs and configurations

Published online by Cambridge University Press:  16 October 2017

NATHAN LINIAL  and
MICHAEL SIMKIN
NATHAN LINIAL
Affiliation:
School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: nati@cs.huji.ac.il)
MICHAEL SIMKIN
Affiliation:
Institute of Mathematics and Federmann Center for the Study of Rationality, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: menahem.simkin@mail.huji.ac.il)
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Abstract

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(nk/(k+1)).

MSC classification

Primary: 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 1 , January 2018 , pp. 69 - 83
Copyright
Copyright © Cambridge University Press 2017 

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