Green's Open Problem 25

For which values of k is the following true: whenever we partition [N] = A_1 ∪ … ∪ A_k, |bigcup^k_i=1 (A_i hat+ A_i)| ≥ 1/10 N?

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	3 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-25,
  title        = {Green's Open Problem 25},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-25}},
  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-25.json)](https://cairn-commons.com/problems/green-25)

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_25. For which values of $k$ is the following true: whenever we partition $[N] = A_{1} \cup \dots \cup A_{k}$ , $⋃_{i = 1}^{k} (A_{i} +^A_{i}) \geq \frac{1}{10} N$ ?

green_25.upper. We conjecture that the best-known upper bound can be lowered.

green_25.lower. We conjecture that the best-known lower bound can be raised.

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[ESS89] Erdős, Pál, András Sárközy, and V. T. Sós. "On a conjecture of Roth and some related problems I." Irregularities of partitions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. 47-59.
[Ru04] Ruzsa, Imre Z. "A problem on restricted sumsets." CONTEMPORARY MATHEMATICS 342 (2004): 245-248.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«25» (3 statements).

theorem green_25 : {k : ℕ → ℕ | ∀ᶠ N in atTop, Property25 (k N) N} = green_25.answer

theorem green_25.upper :
    -- Ensure k is a valid partition size
    (∀ᶠ N in atTop, 1 ≤ green_25.upper.answer N ∧ green_25.upper.answer N ≤ N) ∧
    (fun N => (green_25.upper.answer N : ℝ)) =o[atTop] bestUpper ∧
    ¬ ∀ᶠ N in atTop, Property25 (green_25.upper.answer N) N

theorem green_25.lower :
    (bestLower =o[atTop] green_25.lower.answer) ∧
    (∀ᶠ N in atTop, 1 ≤ green_25.lower.answer N ∧ green_25.lower.answer N ≤ N) ∧
    ∀ k : ℕ → ℕ,
      (∀ᶠ N in atTop, 1 ≤ k N ∧ k N ≤ N) ∧
      ((fun N => (k N : ℝ)) ≪ green_25.lower.answer) →
      ∀ᶠ N in atTop, Property25 (k N) N

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.