Ben Green's Open Problem 1

Let A be a set of n positive integers. Does A contain a sum-free set of size at least frac n 3 + Ω(n), where Ω(n) → ∞ as n → ∞?

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-1,
  title        = {Ben Green's Open Problem 1},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-1}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac{n}{3} + Ω (n)$ , where $Ω (n) \to \infty$ as $n \to \infty$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«1». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_1 : answer(sorry) ↔ ∃ Ω : ℕ → ℝ, atTop.Tendsto Ω atTop ∧
     ∀ n, ∀ (A : Finset ℕ), (∀ a ∈ A, 0 < a) → A.card = n →
     ∃ (S : Finset ℕ), S ⊆ A ∧ IsSumFree (S : Set ℕ) ∧ ((n : ℝ) / 3) + Ω n ≤ S.card

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.

References

[Ben Green's Open Problem 1](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.1 Problem 1)

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.