Erdős (Erdos) Problem #1070: status in 2026

Erdős Problem #1070

Let f(n) be maximal such that, given any n points in ℝ^2, there exist f(n) points such that no two are distance 1 apart. Estimate f(n). In particular, is it true that f(n)≥ n/4? This is the first part, a problem of Erdős [Er87b, p. 171]. Since f(n)=Θ(n) is already known, we ask for lim f(n)/n.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-10-05).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1066 as open (status updated 1 October 2025).
erdosproblems.com lists Erdős problem #1070 as open (status updated 5 October 2025).
Formal Conjectures marks the Lean statement erdos_1070.parts.i as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1066 on erdosproblems.com	open 2025-10-01	erdosproblems.com
Status of Erdős problem #1070 on erdosproblems.com	open 2025-10-05	erdosproblems.com
Formal Conjectures: status of `erdos_1070.parts.i`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 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-1070,
  title        = {Erdős Problem #1070},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1070}},
  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

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 $f (n)$ be maximal such that, given any $n$ points in $R^{2}$ , there exist $f (n)$ points such that no two are distance $1$ apart. Estimate $f (n)$ . In particular, is it true that $f (n) \geq n /4$ ?

This is the first part, a problem of Erdős [Er87b, p. 171]. Since $f (n) = Θ (n)$ is already known, we ask for $lim f (n) / n$ . The limit exists by Fekete's lemma, since placing two configurations far apart shows $f (a + b) \leq f (a) + f (b)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1070».

theorem erdos_1070.parts.i :
    Tendsto (fun n : ℕ ↦ (f n : ℝ) / n) atTop (𝓝 erdos_1070.parts.i.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/1070. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1070
[DuVa26] Dúcz, Á. and Varga, D., *A unit-distance graph in the plane with independence ratio below 1/4*. arXiv:2606.28157 (2026).
[Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. PDF
[MRVZ23] Matolcsi, M., Ruzsa, I. Z., Varga, D. and Zsámboki, P., *The fractional chromatic number of the plane is at least $4$ *. arXiv:2311.10069 (2023).

If we also insist that no two points are distance $< 1$ apart, we get erdosproblems.com/1066.

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.