Ben Green's Open Problem 18

Suppose that G is a finite group, and let A ⊂ G × G be a subset of density α. Is it true that there are ≫_α |G|^3 triples x, y, g such that (x, y), (gx, y), (x, gy) all lie in A? Note: A is taken as α-dense, i.e. |A| ≥ α |G|^2 [Au16, Question 2]

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-18,
  title        = {Ben Green's Open Problem 18},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-18}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $α$ . Is it true that there are $≫_{α} ∣ G ∣^{3}$ triples $x, y, g$ such that $(x, y), (g x, y), (x, g y)$ all lie in $A$ ?

Note: A is taken as $α$ -dense, i.e. $∣ A ∣ \geq α ∣ G ∣^{2}$ [Au16, Question 2]

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«18». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_18 : answer(sorry) ↔
    ∀ α > 0, ∃ c > 0, ∃ m₀ : ℕ,
      ∀ (G : Type*) [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)),
      Fintype.card G ≥ m₀ →
      (A.card : ℝ) ≥ α * (Fintype.card G) ^ 2 →
      (numNaiveCorners A : ℝ) ≥ c * (Fintype.card G) ^ 3

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

[Gr26] Ben Green's Open Problem 18
[Au16] Austin, Tim. "Ajtai–Szemerédi theorems over quasirandom groups." Recent trends in combinatorics. Cham: Springer International Publishing, 2016. 453-484.
[So13] Solymosi, Jozsef. "Roth-type theorems in finite groups." European Journal of Combinatorics 34.8 (2013): 1454-1458.
[Go01] Gowers, William T. "A new proof of Szemerédi's theorem." Geometric & Functional Analysis GAFA 11.3 (2001): 465-588.

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.