Universality of random permutations

TL;DR

This paper advances understanding of the universality of random permutations by showing that permutations of length proportional to $k^2 \, \log \log k$ are typically $k$-universal, containing all patterns of length $k$.

Contribution

It provides the first significant progress towards Alon's conjecture by establishing a new upper bound on permutation length needed for $k$-universality.

Findings

Proves that $n = 2000 k^2 \log \log k$ suffices for $k$-universality.

Improves previous bounds from $O(k^2 \log k)$ to a lower order involving $\log \log k$.

Supports the conjecture that random permutations are highly universal for large $n$.

Abstract

It is a classical fact that for any $\varepsilon > 0$, a random permutation of length $n = (1 + \varepsilon) k^2 / 4$ typically contains a monotone subsequence of length $k$. As a far-reaching generalization, Alon conjectured that a random permutation of this same length $n$ is typically $k$-universal, meaning that it simultaneously contains every pattern of length $k$. He also made the simple observation that for $n = O(k^2 \log k)$, a random length-$n$ permutation is typically $k$-universal. We make the first significant progress towards Alon's conjecture by showing that $n = 2000 k^2 \log \log k$ suffices.