Ben Green's Open Problem 21

Suppose that a_1, …, a_k are integers which do not satisfy Rado's condition: thus if Σ_i ∈ I a_i = 0 then I = ∅. It then follows from Rado's theorem that the equation a_1x_1 + ⋯ + a_kx_k = 0 is not partition regular.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_21 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_21`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	4 open, 1 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. 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-21,
  title        = {Ben Green's Open Problem 21},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-21}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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Disputed

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On the literature board

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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

Suppose that $a_{1}, \dots, a_{k}$ are integers which do not satisfy Rado's condition: thus if $\sum_{i \in I} a_{i} = 0$ then $I = \emptyset$ . It then follows from Rado's theorem that the equation $a_{1} x_{1} + \dots + a_{k} x_{k} = 0$ is not partition regular. Write $c (a_{1}, \dots, a_{k})$ for the least number of colours required in order to colour $N$ so that there is no monochromatic solution to $a_{1} x_{1} + \dots + a_{k} x_{k} = 0$ . Is $c (a_{1}, \dots, a_{k})$ bounded in terms of $k$ only?

This problem, which is known as Rado's boundedness conjecture, dates back to 1933 [Ra33]. It is open for all $k \geq 4$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«21».

theorem green_21 : ∃ B : ℕ → ℕ, ∀ (k : ℕ) (a : Fin k → ℤ),
    ¬ RadoCondition a → minColours a ≤ B k

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_21.variants.fox_kleitman_sharp — Green [Gr24] is not sure that the constant 24 of [FoKl06] is sharp, and remarks that it might be interesting to determine the sharp constant.
green_21.variants.fox_kleitman_modular — A question [FoKl06, Conjecture 5] of Fox and Kleitman, which they call a 'modular analogue' of Rado's Boundedness Conjecture.
green_21.variants.milicevic — Milićević [ElJo23, Conjecture 11.1] conjectures the following 2-adic variant. For any k ∈ ℕ, there exists K = K(k) such that the following is true.

References

[Gr24] Green, Ben. "100 open problems." (2024).
[Ra33] Rado, Richard, Studien zur Kombinatorik. Math. Zeit. 36 (1933), 424-480.
[FoKl06] Fox, Jacob and Kleitman, Daniel, On Rado's boundedness conjecture. J. Combin. Theory Ser. A 113 (2006), no. 1, 84-100.
[ElJo23] Ellis, David and Johnson, Robert (editors), *A collection of open problems in celebration of Imre Leader's 60th birthday*. arXiv preprint arXiv:2310.18163 (2023).

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.