Monotone Subsequences in High-Dimensional Permutations

TL;DR

This paper extends the Erdős-Szekeres theorem to high-dimensional permutations, establishing bounds on the length of monotone subsequences and analyzing their typical size in random permutations.

Contribution

It proves a high-dimensional analogue of the Erdős-Szekeres theorem and characterizes the typical length of monotone subsequences in random high-dimensional permutations.

Findings

Monotone subsequences of length _{k}(\u221a n) exist in all high-dimensional permutations.

Longest monotone subsequences in random permutations are _{k}(n^{k/(k+1)}) asymptotically almost surely.

The bounds are tight, confirming the theoretical limits for high-dimensional cases.

Abstract

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erd\H{o}s-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of length $\Omega_{k}\left(\sqrt{n}\right)$, 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 $\Theta_{k}\left(n^{\frac{k}{k+1}}\right)$.