Green's Open Problem 87

Let p(k) be the limit, as n → ∞, of the probability that a random permutation of [n] preserves some set of size k. Is p(k) a decreasing function of k? This is a conjecture of Peter Cameron (2010), see [BW15].

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 2 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-green-87,
  title        = {Green's Open Problem 87},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-87}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

green_87.decreasing. Let $p (k)$ be the limit, as $n \to \infty$ , of the probability that a random permutation of $[n]$ preserves some set of size $k$ . Is $p (k)$ a decreasing function of $k$ ?

This is a conjecture of Peter Cameron (2010), see [BW15]. The statement is taken in the strict sense, as in [BW15]: they verified $p (k) > p (k + 1)$ for $k < 30$ , and the values computed numerically in [Ve26] satisfy it for $k < 121$ . Note $p (0) = 1$ , since the empty set is always fixed, and $p (1) = 1 - 1/ e$ . The limit $p$ is quantified inside the statement; it exists and is unique by green_87.variants.exists_tendsto.

green_87.asymptotic. Is $p (k) = (C + o (1)) k^{- δ} (lo g k)^{- 3/2}$ for some absolute constant $C$ ?

Green and Sawhney [GS26] proved $p (k) = (1 + o (1)) f ({lo g_{2} k}) k^{- δ} (lo g k)^{- 3/2}$ for an explicit smooth, positive $f$ on $R / Z$ with $max f / min f < 1 + 2 \cdot 1 0^{- 7}$ . The question is therefore equivalent to whether $f$ is constant; they conjecture that it is not, so the expected answer is no, but this has not been proved.

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[BW15] Britnell, John R., and Mark Wildon. "Computing derangement probabilities of the symmetric group acting on k-sets." arXiv preprint arXiv:1511.04106 (2015).
[EFG16] Eberhard, Sean, Kevin Ford, and Ben Green. "Permutations fixing a k-set." International Mathematics Research Notices 2016.21 (2016): 6713-6731. arXiv:1507.04465.
[GS26] Green, Ben, and Mehtaab Sawhney. "The proportion of permutations fixing a k-set." arXiv preprint arXiv:2604.28116 (2026).
[Ve26] Veljjanoski, Daniel. "p(k) for permutations fixing a k-set: values and checks for k ≤ 121." https://github.com/veljjanoski/green87-fixing-k-set (2026).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«87» (2 statements).

theorem green_87.decreasing :
    ∀ p : ℕ → ℝ, (∀ k, Tendsto (fun n ↦ i n k) atTop (𝓝 (p k))) → StrictAnti p

theorem green_87.asymptotic :
    ∀ p : ℕ → ℝ, (∀ k, Tendsto (fun n ↦ i n k) atTop (𝓝 (p k))) →
      ∃ C : ℝ, 0 < C ∧
        p ~[atTop] fun k : ℕ ↦ C * (k : ℝ) ^ (-δ) * (Real.log k) ^ (-(3 / 2 : ℝ))

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.