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Level A · Machine-checkable Hard Analysis P-green-85

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?

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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
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Disputed
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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

Suppose that is an open subset of with measure . Are there four points in determining an axis-parallel rectangle with area ?

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

theorem green_85 :
  answer(sorry) ↔ ∃ 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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.