Quadratic-size Random Permutations are Universal

A permutation is $n$-universal if it contains every permutation of length $n$ as a pattern. We prove that a uniformly random permutation of length $32n^2$ is $n$-universal with probability at least $1-\exp (-cn\mathrm{/}\log n)$, for an absolute constant $c>0$ and all sufficiently large $n$. This removes a $\log \log n$ factor from an earlier result of He and Kwan~\cite{He-Kwan} and resolves an open problem of Alon~\cite{Alon2016}. The proof was found by GPT-6 Pro.