Green's Open Problem 85

Suppose that A is an open subset of [0, 1]^2 with measure α. Are there four points in A determining an axis-parallel rectangle with area gt c α^2?

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_85 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_85`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 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-85,
  title        = {Green's Open Problem 85},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-85}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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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 $A$ is an open subset of $[0, 1]^{2}$ with measure $α$ . Are there four points in $A$ determining an axis-parallel rectangle with area $> c α^{2}$ ?

Carbery’s rectangle problem

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[CCW99] Carbery, Anthony, Michael Christ, and James Wright. "Multidimensional van der Corput and sublevel set estimates." Journal of the American Mathematical Society 12.4 (1999): 981-1015 Section 6.
[Ke00] Keleti, Tamás. "Density and covering properties of intervals of ℝn." Mathematika 47.1-2 (2000): 229-242.
[KKM02] Katz, Nets Hawk, Elliot Krop, and Mauro Maggioni. "Remarks on the box problem." Mathematical Research Letters 9.4 (2002): 515-520.
[Mu02] Mubayi, Dhruv. "Some exact results and new asymptotics for hypergraph Turán numbers." Combinatorics, Probability and Computing 11.3 (2002): 299-309 Conjecture 1.4.
[CPZ20] Conlon, David, Cosmin Pohoata, and Dmitriy Zakharov. "Random multilinear maps and the Erd\H {o} s box problem." arXiv preprint arXiv:2011.09024 (2020).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«85».

theorem green_85 :
  ∃ c > 0, ∀ A : Set (ℝ × ℝ),
    IsOpen A →
    A ⊆ Icc 0 1 ×ˢ Icc 0 1 →
    A.Nonempty →
    let α := (volume A).toReal
    ∃ x₁ x₂ y₁ y₂,
      {(x₁, y₁), (x₂, y₁), (x₂, y₂), (x₁, y₂)} ⊆ A ∧
      c * α ^ 2 ≤ |x₁ - x₂| * |y₁ - y₂|

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.

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.