Green's Open Problem 36

Do the following exist, for arbitrarily large n? An abelian group H with |H| = n^2+o(1), together with subsets A_1, ..., A_n, B_1, ..., B_n satisfying |A_i||B_i| ≥ n^2-o(1) and |A_i + B_i| = |A_i||B_i|, such that the sets A_i + B_i are disjoint from the sets A_j + B_k (j ≠ k)?

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-36,
  title        = {Green's Open Problem 36},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-36}},
  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

Do the following exist, for arbitrarily large $n$ ? An abelian group $H$ with $∣ H ∣ = n^{2 + o (1)}$ , together with subsets $A_{1}, ..., A_{n}, B_{1}, ..., B_{n}$ satisfying $∣ A_{i} ∣∣ B_{i} ∣ \geq n^{2 - o (1)}$ and $∣ A_{i} + B_{i} ∣ = ∣ A_{i} ∣∣ B_{i} ∣$ , such that the sets $A_{i} + B_{i}$ are disjoint from the sets $A_{j} + B_{k}$ ( $j \neq = k$ )?

NOTE: according to [CKS05, 4.1], the conditions should be $A_{i} + B_{j}$ disjoint from $A_{j} + B_{k}$ for $i \neq = k$ . See green_36.variants.cks05.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«36». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_36 :
    answer(sorry) ↔
      ∀ ε > (0 : ℝ), ∃ᶠ n in atTop,
        ∃ (H : Type) (_ : AddCommGroup H) (_ : Finite H) (A B : Fin n → Finset H),
          (n : ℝ) ^ (2 - ε) ≤ Nat.card H ∧ Nat.card H ≤ (n : ℝ) ^ (2 + ε) ∧
          (∀ i, (n : ℝ) ^ (2 - ε) ≤ (A i).card * (B i).card) ∧
          Green36Property A B

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.

Variants

green_36.variants.cks05 — Variant using the exact simultaneous double product property from [CKS05, 4.1].

References

[Gr24] Green's Open Problems #36
[CKS05] Cohn, H., Kleinberg, R., Szegedy, B., and Umans, C. "Group-theoretic Algorithms for Matrix Multiplication" (Problem 4.7)

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.