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Level A · Machine-checkable Hard Combinatorics P-green-25

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?

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.

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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 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

green_25. For which values of is the following true: whenever we partition , ?

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). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_25 : {k : ℕ → ℕ | ∀ᶠ N in atTop, Property25 (k N) N} = answer(sorry)
theorem green_25.upper :
    let ans := (answer(sorry) : ℕ → ℕ)
    (∀ᶠ N in atTop, 1 ≤ ans N ∧ ans N ≤ N) ∧ -- Ensure k is a valid partition size
    (fun N => (ans N : ℝ)) =o[atTop] bestUpper ∧
    ¬ ∀ᶠ N in atTop, Property25 (ans N) N
theorem green_25.lower :
    let ans := (answer(sorry) : ℕ → ℝ)
    (bestLower =o[atTop] ans) ∧
    (∀ᶠ N in atTop, 1 ≤ ans N ∧ ans N ≤ N) ∧
    ∀ k : ℕ → ℕ,
      (∀ᶠ N in atTop, 1 ≤ k N ∧ k N ≤ N) ∧
      ((fun N => (k N : ℝ)) ≪ ans) →
      ∀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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.