Green's Open Problem 12

Let G be an abelian group of size N, and suppose that A ⊂ G has density α. Are there at least α^15 N^10 tuples (x_1, …, x_5, y_1, …, y_5) ∈ G^10 such that x_i + y_j ∈ A whenever j ∈ i, i+1, i+2? Note: We interpret indices modulo 5.

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-12,
  title        = {Green's Open Problem 12},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-12}},
  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 $G$ be an abelian group of size $N$ , and suppose that $A \subset G$ has density $α$ . Are there at least $α^{15} N^{10}$ tuples $(x_{1}, \dots, x_{5}, y_{1}, \dots, y_{5}) \in G^{10}$ such that $x_{i} + y_{j} \in A$ whenever $j \in {i, i + 1, i + 2}$ ?

Note: We interpret indices modulo 5.

References:

[Gr24] Green, Ben. "100 open problems." (2024).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«12». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_12 : answer(sorry) ↔
    ∀ {G : Type*} [AddCommGroup G] [Fintype G] [DecidableEq G],
    ∀ (A : Finset G),
    let N := Fintype.card G
    let α := (A.card : ℝ) / N
    let valid_tuples : Finset ((Fin 5 → G) × (Fin 5 → G)) := Finset.univ.filter (fun t =>
      ∀ i : Fin 5, ∀ j ∈ ({i, i + 1, i + 2} : Finset (Fin 5)), t.1 i + t.2 j ∈ A)
    (valid_tuples.card : ℝ) ≥ α ^ 15 * (N : ℝ) ^ 10

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.

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.