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Level A · Machine-checkable Hard Combinatorics P-green-40

Ben Green's Open Problem 40

Does f(r) → ∞? [Gr24]

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-40,
  title        = {Ben Green's Open Problem 40},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-40}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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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

green_40. Does ? [Gr24]

green_40.f_eq_one_for_all. The possibility that f(r) = 1 for all r has not been ruled out [Gr24]

green_40.f_two_eq_one. It is not known whether f(2) = 1 [Gr24]

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«40» (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_40 : answer(sorry) ↔ Tendsto f atTop (𝓝 ⊤)
theorem green_40.f_eq_one_for_all : answer(sorry) ↔ ∀ r, f r = 1
theorem green_40.f_two_eq_one : answer(sorry) ↔ f 2 = 1

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.

Variants

  • green_40.variants.arbitrary_subsets — Does tildef(r) → ∞? [Gr24]
  • green_40.variants.all_n — Does f_all(r) → ∞? [Gr24] The target filter is 𝓝 ⊤, as in green_40 and green_40.variants.arbitrary_subsets.

References

  • [Gr24] Ben Green's Open Problem 40
  • [Da90] Davydov, Alexander Abramovich. "Construction of linear covering codes." Problemy Peredachi Informatsii 26.4 (1990): 38-55.
  • [CHL97] Cohen, G., Honkala, I., Litsyn, S., & Lobstein, A. (1997). Covering codes (Vol. 54). Elsevier.
  • [St94] R. Struik, Covering codes, PhD Thesis, Eindhoven University of Technology, the Netherlands, 106 pp, 1994.

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.