Ben Green's Open Problem 5

Which finite groups have the smallest biggest product-free sets? We formalise this as: determine the supremum of exponents α such that every nontrivial finite group of order n contains a product-free set of size ≥ c n^α for some absolute constant c > 0.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_5 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_5`	research open	Formal Conjectures
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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-green-5,
  title        = {Ben Green's Open Problem 5},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-5}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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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

Which finite groups have the smallest biggest product-free sets?

We formalise this as: determine the supremum of exponents $α$ such that every nontrivial finite group of order $n$ contains a product-free set of size $\geq c n^{α}$ for some absolute constant $c > 0$ . (The trivial group is excluded since its only product-free subset is empty.) Kedlaya [Ke97] showed that $α = 11/14$ is admissible, and Green suggests this exponent may well be sharp; the candidate extremal family is the Ree groups $^{2} G_{2} (q)$ , $q = 3^{2 m + 1}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«5».

theorem green_5 :
    IsLUB {α : ℝ | ∃ c > (0 : ℝ), ∀ (G : Type) [Group G] [Fintype G], Nontrivial G →
      ∃ S : Finset G, IsProductFree (S : Set G) ∧
        c * (Fintype.card G : ℝ) ^ α ≤ (S.card : ℝ)}
      green_5.answer

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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.

Variants

green_5.variants.sl_two — A good model problem would be to determine the largest product-free subsets of SL_2(𝔽_p).

References

[Gr24] Green, Ben. "100 open problems." (2024).
[BaSo85] Babai L, Sós VT. Sidon sets in groups and induced subgraphs of Cayley graphs. European Journal of Combinatorics. 1985 Jun 1;6(2):101-14.
[Ke97] Kedlaya, K. S., Large product-free subsets of finite groups, J. Combin. Theory Ser. A 77 (1997), no. 2, 339–343.
[Ke09] Kedlaya, K. S., Product-free subsets of groups, then and now, Contemp. Math., 479, American Mathematical Society, Providence, RI, 2009, 169–177.
[Go08] Gowers, W. T., Quasirandom groups, Combin. Probab. Comput. 17 (2008), no. 3, 363–387.

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.