---
id: "2026-10-09-random-permutations-universal-alon-gpt-6-pro"
url: "https://postcutoff.com/e/2026-10-09-random-permutations-universal-alon-gpt-6-pro/"
as_of: "2026-10-12T07:22:00+02:00"
date: "2026-10-09"
date_precision: day
category: science
importance: 3
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 1
editor: Adam Bicz
human_review: null
version: null
---

As of: 2026-10-12 07:22 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz.
Canonical page: https://postcutoff.com/e/2026-10-09-random-permutations-universal-alon-gpt-6-pro/

# Random permutations of quadratic length contain every pattern

Full title: Random permutations of quadratic length contain every pattern: Alon's 1000n² problem solved, 'the proof was found by GPT-6 Pro' (Chen, Wang)

On 9 Oct 2026 Yu Chen and Yiting Wang posted arXiv 2610.13530: a uniformly random permutation of length 32n² contains every permutation of length n as a pattern with probability at least 1 − exp(−cn/log n). This removes the log log n factor from He and Kwan's bound and solves Noga Alon's problem of showing that 1000n² suffices. The abstract ends: "The proof was found by GPT-6 Pro", and the declaration adds "The authors do not claim credit for the proof ideas." Alon's sharper conjecture, with constant 1/4, remains open. Unrefereed preprint.

## Key facts
- Paper: 'Quadratic-size Random Permutations are Universal', arXiv 2610.13530 [math.CO], 9 Oct 2026; Yu Chen (Beijing Institute of Technology) and Yiting Wang (supported by the ERC grant RANDSTRUCT)
- Theorem 1.1: there are absolute constants c > 0 and n₀ such that for n ≥ n₀ a uniformly random permutation of length 32n² fails to be n-universal with probability at most exp(−cn/log n)
- Background (paper): Alon conjectured that length (1 + ε)n²/4 suffices, observed that O(n² log n) does, 'and posed the weaker problem of proving that 1000n² suffices'; He and Kwan proved 2000n² log log n. Length below about n²/4 cannot work, because of the longest increasing subsequence
- What remains open (paper): 'the sharp constant 1/4 predicted by Alon's conjecture remains open'; the authors sketch how to lower 32 to any constant above 1 and why 1 is a barrier for the method
- Declaration on the usage of generative AI: 'The proof was found by GPT-6 Pro. The authors do not claim credit for the proof ideas. They have verified and digested the proof and take full responsibility for the correctness of the manuscript.'
- Not found as of 12 Oct 2026: reactions from Alon, He or Kwan, or other expert comment

## What happened

A permutation is n-universal if every ordering of n items appears inside it as a pattern. Any universal permutation needs length
of order n², and Noga Alon asked whether a random permutation of length a constant times n² is already universal. Xiaoyu He and
Matthew Kwan came within a log log n factor. Chen and Wang's paper removes that factor, so the order of magnitude is now known.

The authors state plainly that the model did the mathematics: GPT-6 Pro found the proof, and their own contribution was to
verify, digest and write it. The prompt and the time taken are not given.

## Why it matters

It is a clean case of a model resolving a problem that specialists had posed and partly solved, with authors who attribute the
ideas to the model in the abstract. The problem is of interest to specialists more than a famous conjecture, and the result is
not sharp.

## What is disputed or not yet verified

- Verification: Unrefereed preprint; proof checked by the authors

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 162 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 101 days after its cutoff
- Gemini 3.8 Flash (training cutoff March 2026): 192 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 131 days after its cutoff

## Sources

1. [arXiv 2610.13530: Quadratic-size Random Permutations are Universal](https://arxiv.org/abs/2610.13530) (arxiv.org, paper)

## Changes

- 2026-10-12 (filed): Created from the arXiv PDF

## Related

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- 2026-10-06: [Optimal bound for the polynomial Littlewood–Offord problem, 'discovered autonomously by GPT-6 Pro' (exposition by Grebennikov)](https://postcutoff.com/e/2026-10-06-polynomial-littlewood-offord-gpt-6-pro/index.md)
- 2026-10-04: [Linear Hadwiger conjecture proved](https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/index.md)