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Level A · Machine-checkable Hard Combinatorics P-green-5

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.

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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

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

We formalise this as: determine the supremum of exponents such that every nontrivial finite group of order contains a product-free set of size for some absolute constant . (The trivial group is excluded since its only product-free subset is empty.) Kedlaya [Ke97] showed that is admissible, and Green suggests this exponent may well be sharp; the candidate extremal family is the Ree groups , .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«5». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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)}
      answer(sorry)

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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.