Ben Green's Open Problem 72

The no-k-in-line problem: For which k > 2 does every N × N grid with N ≥ k contain a set of (k - 1) N points with no k on a line, so that AllowedSetSize k N is the pigeonhole bound (k - 1) N?

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
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.

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Cite

@misc{cairn-green-72,
  title        = {Ben Green's Open Problem 72},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-72}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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On the literature board

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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

NoKInLine. The no-k-in-line problem: For which $k > 2$ does every $N \times N$ grid with $N \geq k$ contain a set of $(k - 1) N$ points with no $k$ on a line, so that AllowedSetSize k N is the pigeonhole bound $(k - 1) N$ ?

[GK2025] proves that every $k > 1 0^{37}$ has this property, which is no_k_in_line_big below, and does not optimise this constant. At $k = 3$ it is the property that Green expects to fail for large $N$ , see green_72.

green_72. Green's Open Problem 72 / No-three-in-line problem: For $N$ sufficiently large, is it impossible to have $2 N$ points in $[N]^{2}$ with no three in a line? Green suspects the answer is yes, and that $(3/2 + o (1)) N$ is optimal.

More commonly known as the no-three-in-line problem.

What is the largest subset of the grid $[N]^{2}$ with no three points in a line? In particular, for $N$ sufficiently large, is it impossible to have a set of size $2 N$ with this property?

The upper bound $2 N$ is the easy half and is allowedSetSize_le below, by pigeonhole on the columns. The open content is whether $2 N$ is attained. Green records that it is for $N$ up to around 50, that $(3/2 + o (1)) N$ points are achievable for arbitrary $N$ , and that his "personal suspicion is that this is optimal". The Wikipedia reference points the same way: Guy and Kelly conjectured $c = 3 2 π^{2} /3 \approx 1.874$ , and after an error in the heuristic was found Guy corrected it to $c = π / 3 \approx 1.814$ . Both are below $2$ , so the expected answer to the question above is yes.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«72» (2 statements).

theorem NoKInLine : NoKInLine.answer = {k | 2 < k ∧ ∀ N, k ≤ N → NoKInLineFor k N}

theorem green_72 : ∀ᶠ N in Filter.atTop, ¬ NoKInLineFor 3 N

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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_72.variants.eventually — Is 2N attained for all sufficiently large N? This is not the negation of green_72.

References

Ben Green's Open Problem 72
Wikipedia
[GK2025] Grebennikov, A. Kwan, M. No $(k + 1)$ -in-line problem for large constant $k$ . https://arxiv.org/abs/2510.17743

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.