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Level A · Machine-checkable Hard Combinatorics P-green-88

Ben Green's Open Problem 88

[Gr24, Problem 88]: Is there an absolute constant δ > 0 such that every pairwise 2-comparable set S ⊆ [N]^3 satisfies |S| ≤ N^2 - δ?

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-88,
  title        = {Ben Green's Open Problem 88},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-88}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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.

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Current state

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The problem

The question

[Gr24, Problem 88]: Is there an absolute constant such that every pairwise -comparable set satisfies ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«88».

theorem green_88 :
    ∃ δ > (0 : ℝ), ∀ (N : ℕ) (S : Finset (Fin 3 → Fin N)),
      (S : Set (Fin 3 → Fin N)).Pairwise IsComparable₂ →
        (S.card : ℝ) ≤ (N : ℝ) ^ (2 - δ)

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.

References

  • [Gr24] [Ben Green's Open Problem 88](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.88)
  • [GoLo21] [The length of an -increasing sequence of -tuples](https://arxiv.org/abs/1609.08688) by W. T. Gowers and J. Long

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.