Problem 1070

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 $f(n)\geq n/4$ ?

Source: erdosproblems.com/1070

Geometry

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Status

Open

No claim settles this problem.

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 (Ducz Varga, 2026) without being adopted; the standing in the frontmatter is derived from the claim pages.

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).

Problem 1070

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Proof claims

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