Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 1070

claims/: The 1 claim page of Problem 1070, one per claimant's result; the problem's standing derives from them.

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

Status. Open: the site labels the problem OPEN (page last edited 22 January 2026), and its proof-claims tab carries one partial proof claim, Ákos Dúcz and Dániel Varga's preprint of 26 June 2026 answering the particular question in the negative, recorded on its claim page without being adopted; the standing in the frontmatter is derived from the claim pages.

Source. erdosproblems.com/1070, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1070, https://www.erdosproblems.com/1070.

References.

[ACMVZ23] G. Ambrus, A. Csiszárik, M. Matolcsi, D. Varga, and P. Zsámboki, The density of planar sets avoiding unit distances. Math. Program. 207 (2024), 303-327; arXiv:2207.14179.
[Cr67] H. T. Croft, Incidence incidents. Eureka (1967), 22-26.
[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.
[MRVZ23] M. Matolcsi, I. Z. Ruzsa, D. Varga, and P. Zsámboki, The fractional chromatic number of the plane is at least $4$ . arXiv:2311.10069 (2023).

Formalization. None built or audited here. Two public Lean developments of the pending claim's Theorem 1 are linked, pinned, on its claim page; formal-conjectures has no statement file for the problem.

Current assessment

Question and standing. The site formulation above asks for the order of $f(n)$ , the largest number of points that every $n$ -point planar set is guaranteed to contain with no two at distance one, equivalently the least independence number of an $n$ -vertex unit-distance graph, and in particular whether $f(n)\ge n/4$ . No claim settles the problem, so it is open. One pending partial claim bears on the particular question: Dúcz and Varga's preprint proves that some finite planar unit-distance graph has independence ratio below $1/4$ , which, if it holds, gives $f(n)<n/4$ for all large $n$ and so a negative answer to the particular question; it is unrefereed and unreviewed, its Lean formalizations have not been built by this corpus, and the claimants call it partial because the estimate of $f(n)$ remains.
Known results. The site records the bounds. Since the independence number times the chromatic number is at least the number of vertices, $f(n)\ge n/\chi$ with $\chi$ the chromatic number of the plane (Problem 508). The Moser spindle gives $f(n)\le\frac27n$ . Larman and Rogers [LaRo72] observed $f(n)\ge m_1n$ , where $m_1$ is the supremum of upper densities of measurable planar sets with no two points at distance one, and Croft's construction [Cr67] gives $m_1\ge0.22936$ ; Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki [ACMVZ23] proved $m_1\le0.247$ , so this route cannot reach $n/4$ . Matolcsi, Ruzsa, Varga and Zsámboki [MRVZ23] proved $f(n)\le(\frac14+o(1))n$ ([[../library/discrete_geometry/matolcsi_2025_fractional_chromatic_number_plane_is_at/theorem_2|Theorem 2]]) and conjectured that $m_1$ equals Croft's bound and that the finitary independence ratio of the plane is $\frac14$ with every finite planar unit-distance graph of fractional chromatic number below $4$ ([[../library/discrete_geometry/matolcsi_2025_fractional_chromatic_number_plane_is_at/conjecture_1|Conjecture 1]]), which gives $f(n)>n/4$ for every $n$ ; the claim above contradicts the second conjecture. The variant with no two points at distance less than one is Problem 1066.
Status search. The search covers the site's page and its proof-claims tab and the arXiv record of the preprint,; the paper is compiled on its source card. No broader literature search is recorded.
Proof coverage and review. None. The paper's certificate and blow-ups have not been independently checked, no own-words proof is held, and no independent review is recorded.

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