Erdős (Erdos) Problem #1066: status in 2026

Erdős Problem #1066

Let G be a graph given by n points in ℝ^2, where any two distinct points are at least distance 1 apart, and we draw an edge between two points if they are distance 1 apart. Let g(n) be maximal such that any such graph always has an independent set on at least g(n) vertices.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-10-01).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1066 as open (status updated 1 October 2025).
Formal Conjectures marks the Lean statement erdos_1066 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1066 on erdosproblems.com	open 2025-10-01	erdosproblems.com
Formal Conjectures: status of `erdos_1066`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 3 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-1066,
  title        = {Erdős Problem #1066},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1066}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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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 $G$ be a graph given by $n$ points in $R^{2}$ , where any two distinct points are at least distance $1$ apart, and we draw an edge between two points if they are distance $1$ apart.

Let $g (n)$ be maximal such that any such graph always has an independent set on at least $g (n)$ vertices. Estimate $g (n)$ , or perhaps $lim \frac{g ( n )}{n}$ .

We ask for $lim g (n) / n$ . The limit exists by Fekete's lemma, since placing two configurations far apart shows $g (a + b) \leq g (a) + g (b)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1066».

theorem erdos_1066 :
    Tendsto (fun n : ℕ ↦ (g n : ℝ) / n) atTop (𝓝 erdos_1066.answer)

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/1066. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_1066.variants.higher_dim — Pollack [Po85] also reports a letter from Erdős which poses the more general problem of, given n points in ℝ^d with minimum distance 1, let g_d(n) be maximal…

References

erdosproblems.com/1066
[Cs98] Csizmadia, G., On the independence number of minimum distance graphs. Discrete Comput. Geom. 20 (1998), 179-187.
[Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
[PaTo96] Pach, J. and Tóth, G., On the independence number of coin graphs. Geombinatorics 6 (1996), 30-33.
[Po85] Pollack, R., Increasing the minimum distance of a set of points. J. Combin. Theory Ser. A 40 (1985), 450.
[Sw02] Swanepoel, K. J., Independence numbers of planar contact graphs. Discrete Comput. Geom. 28 (2002), 649-670.

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.