Ben Green's Open Problem 40

Does f(r) → ∞? [Gr24]

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	5 open, 5 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.

Try it with your chatbot Submit a claim Follow (RSS)

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 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Fgreen-40.json)](https://cairn-commons.com/problems/green-40)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

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_40. Does $f (r) \to \infty$ ? [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).

theorem green_40 : Tendsto f atTop (𝓝 ⊤)

theorem green_40.f_eq_one_for_all : ∀ r, f r = 1

theorem green_40.f_two_eq_one : 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.