Ben Green's Open Problem 75

Let X ⊂ ℝ^2 be a set of n points. Does there exist a line ℓ through at least two points of X such that the numbers of points on either side of ℓ differ by at most 100? This is a form of the Kupitz-Perles conjecture [Gr24]. We take n ≥ 2, since a line through two points of X needs two points.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_75 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_75`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 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-75,
  title        = {Ben Green's Open Problem 75},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-75}},
  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 $X \subset R^{2}$ be a set of $n$ points. Does there exist a line $ℓ$ through at least two points of $X$ such that the numbers of points on either side of $ℓ$ differ by at most $100$ ?

This is a form of the Kupitz-Perles conjecture [Gr24]. We take $n \geq 2$ , since a line through two points of $X$ needs two points. If all points of $X$ are collinear, the line through them has no points on either side.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«75».

theorem green_75 :
    ∀ X : Finset (ℝ × ℝ), 2 ≤ #X → ∃ p ∈ X, ∃ q ∈ X, p ≠ q ∧ imbalance X p q ≤ 100

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_75.variants.exists_constant — Is there a constant C such that every finite set X ⊂ ℝ^2 with at least two points has a line through two points of X whose sides differ by at most C?

References

[Gr24] Green, Ben. "100 open problems." (2024).
[CL25] Conlon, D. and Lim, J., Everywhere unbalanced configurations. Adv. Math. 480 (2025), Paper No. 110445. arXiv:2308.02466
[Pi03] Pinchasi, R., Lines with many points on both sides. Discrete Comput. Geom. 30 (2003), 415-435. doi:10.1007/s00454-003-2826-8

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.