Green's Open Problem 24

If A is a set of n integers, what is the maximum number of affine translates of the set lbrace 0,1,3 rbrace that A can contain? Conjectured in [Aa19] p.579: (1/3 + o(1)) n^2.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 3 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.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-green-24,
  title        = {Green's Open Problem 24},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-24}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Fgreen-24.json)](https://cairn-commons.com/problems/green-24)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

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

green_24. If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set ${0, 1, 3}$ that $A$ can contain?

Conjectured in [Aa19] p.579: $(\frac{1}{3} + o (1)) n^{2}$ .

conjecture. Conjecture p.579 in [Aa19]: $(\frac{1}{3} + o (1)) n^{2}$ .

References:

Green, Ben. "100 open problems." (2024).
[Aa19] Aaronson, James. "Maximising the number of solutions to a linear equation in a set of integers." Bulletin of the London Mathematical Society 51.4 (2019): 577-594.
[HaL28] Hardy, G. H., and J. E. Littlewood. "Notes on the theory of series (VIII): an inequality." Journal of the London Mathematical Society 1.2 (1928): 105-110.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«24» (2 statements).

theorem green_24 : ∀ n, max013AffineTranslates n = green_24.answer n

theorem conjecture : gamma = 1/3

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.