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Updated Oct 8

Problem 232

claims/: The 1 claim page of Problem 232, one per claimant's result; the problem's standing derives from them.

Statement. For $A\subset \mathbb{R}^2$ we define the upper density as

\overline{\delta}(A)=\limsup_{R\to \infty}\frac{\lambda(A \cap B_R)}{\lambda(B_R)},

where $\lambda$ is the Lebesgue measure and $B_R$ is the ball of radius $R$ .

Estimate

m_1=\sup \overline{\delta}(A),

where $A$ ranges over all measurable subsets of $\mathbb{R}^2$ without two points distance $1$ apart. In particular, is $m_1\leq 1/4$ ?

Status. Proved. The site's label answers the particular question: Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki proved $m_1\le0.247<1/4$ , which answers the site's question and proves Erdős's conjecture $m_1<1/4$ [Er85], read with the circle's area as the normalization. The conjecture as printed (p. 4) divides the measure by $R^2$ rather than by the area $\pi R^2$ , and in that form it is false. The estimate of $m_1$ itself remains between Croft's lower bound $0.22936$ and this upper bound. The result is recorded in claims/ as an accepted claim.

Source. erdosproblems.com/232, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #232, https://www.erdosproblems.com/232.

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, doi:10.1007/s10107-023-02012-9; arXiv:2207.14179.
[Cr67] H. T. Croft, Incidence incidents, Eureka 30 (1967), 22–26.
[Er85] P. Erdős, Problems and results in combinatorial geometry, in Discrete geometry and convexity (New York, 1982), Ann. New York Acad. Sci. 440 (1985), 1–11.
[Mo66] L. Moser, Poorly formulated unsolved problems of combinatorial geometry. Mimeographed (1966).

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (14)

The density bound 0.247 for planar sets avoiding unit distances problems/distance_problems/E0232/claims Distance Problems discrete_geometry/decorte_2020_complete_positivity_distance_avoiding_sets Theorem 1.1 (p. 3): completely positive type makes the sphere and Euclidean relaxations exact Theorem 6.3 (p. 26): the completely positive program gives the independence density of G(R^n, D) for closed D discrete_geometry/ducz_2026_unit_distance_graph_plane_independence_ratio Conjecture 1 (p. 3): f(n)/n tends to the measurable density m_1 Corollary 3 (p. 1): measurable planar sets avoiding distance 1 have density below 1/4 Distance Problems distance_problems/erdos_1985_problems_results_combinatorial_geometry Conjecture, p. 4: the limit in L. Moser's problem (10) is less than 1/4 distance_problems/graham_2010_open_problems_euclidean_ramsey_theory Unit distance survey (pp. 3-4): chromatic numbers of the plane, of 3- and 4-space and of rational space, and dense sets avoiding distance one

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