Erdős (Erdos) Problem #588: status in 2026

Erdős Problem #588

Let f_k(n) be minimal such that if n points in ℝ^2 have no k+1 points on a line then there must be at most f_k(n) many lines containing at least k points. Is it true that f_k(n) = o(n^2) for k ≥ 4? The case k = 4 is Erdős Problem 101.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $100.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #588 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_588 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #588 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_588`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 7 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), 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-erdos-588,
  title        = {Erdős Problem #588},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-588}},
  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

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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 $f_{k} (n)$ be minimal such that if $n$ points in $R^{2}$ have no $k + 1$ points on a line then there must be at most $f_{k} (n)$ many lines containing at least $k$ points. Is it true that $f_{k} (n) = o (n^{2})$ for $k \geq 4$ ?

The case $k = 4$ is Erdős Problem 101. The restriction $k \geq 4$ is needed: see Erdos588.erdos_588.variants.three_not_isLittleO.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«588».

theorem erdos_588 :
    ∀ k : ℕ, 4 ≤ k →
      (fun n : ℕ ↦ (numLinesWithPointsMax k n : ℝ)) =o[atTop] (fun n : ℕ ↦ (n : ℝ) ^ 2)

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 — check erdosproblems.com/588. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/588
[BGS74] Burr, Stefan A. and Grünbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.
[Er84] Erdős, P., Research problems. Period. Math. Hungar. (1984), 101-103.
[FuPa84] Füredi, Z. and Palásti, I., *Arrangements of lines with a large number of triangles*. Proc. Amer. Math. Soc. (1984), 561-566.
[Gr76] Grünbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo I (1976), 451-468.
[Ka63] Kárteszi, F., Sylvester egy tételéről és Erdős egy sejtéséről. Matematikai Lapok (1963), 3-10.
[SoSt13] Solymosi, József and Stojaković, Miloš, *Many collinear $k$ -tuples with no $k + 1$ collinear points*. Discrete Comput. Geom. (2013), 811-820.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.