Ben Green's Open Problem 76

Let A be a set of n points in the plane. Can one select a set A' of n/2 points, with the property that any axis-parallel rectangle containing 1000 points of A contains at least one point of A'?

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_76 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_76`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 open, 2 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-76,
  title        = {Ben Green's Open Problem 76},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-76}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

Let $A$ be a set of $n$ points in the plane. Can one select a set $A^{'}$ of $n /2$ points, with the property that any axis-parallel rectangle containing $1000$ points of $A$ contains at least one point of $A^{'}$ ?

We read "a set of $n /2$ points" as at most $⌊ n /2 ⌋$ points, and "containing $1000$ points" as containing at least $1000$ points. The points of $A^{'}$ need not lie in $A$ . If $A^{'} \subseteq A$ is required, then the answer is no; see green_76.variants.subset.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«76».

theorem green_76 :
    ∀ A : Finset (ℝ × ℝ), ∃ A', #A' ≤ #A / 2 ∧ HitsRichRectangles 1000 A A'

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.

Variants

green_76.variants.exists_constant — Is there a constant k such that the answer to green_76 is yes with 1000 replaced by k?
green_76.variants.weak_net — Do weak ε-nets of size O(1/ε) exist for axis-parallel rectangles in the plane?

References

[Gr24] Green, Ben. "100 open problems." (2024).
[AES10] Aronov, Boris, Esther Ezra, and Micha Sharir. "Small-size $ε$ -nets for axis-parallel rectangles and boxes." SIAM Journal on Computing 39.7 (2010): 3248-3282. doi:10.1137/090762968
[PT11] Pach, János. "Boxes and the art of $ε$ -net maintenance." Invited problems, 2nd Emléktábla Workshop (2011), Problem 5 (Pach and Tardos). PDF
[PT13] Pach, János, and Gábor Tardos. "Tight lower bounds for the size of epsilon-nets." Journal of the American Mathematical Society 26.3 (2013): 645-658. doi:10.1090/S0894-0347-2012-00759-0, arXiv:1012.1240

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.