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Updated Oct 8

Problem 223

claims/: The 6 claim pages of Problem 223, one per claimant's result; the problem's standing derives from them.

Statement. Let $d\geq 2$ and $n\geq 2$ . Let $f_d(n)$ be maximal such that there exists some set of $n$ points $A\subseteq \mathbb{R}^d$ , with diameter $1$ , in which the distance 1 occurs between $f_d(n)$ many pairs of points in $A$ . Estimate $f_d(n)$ .

Status. Solved. The site credits $f_2(n)=n$ for $n\ge3$ (Hopf and Pannwitz), $f_3(n)=2n-2$ for $n\ge4$ (Grünbaum, Heppes and Straszewicz, independently), and for $d\ge4$ the asymptotic $f_d(n)=(\frac{p-1}{2p}+o(1))n^2$ with $p=\lfloor d/2\rfloor$ (Erdős), with the exact value and the extremal sets for all $n\ge n_0(d)$ (Swanepoel). The three ranges of dimensions are the problem's parts, listed in the frontmatter as plane, space and higher_dimensions; each result is recorded in claims/ as an accepted partial claim settling its part, and the problem's standing derives from the six claims together.

Source. erdosproblems.com/223, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #223, https://www.erdosproblems.com/223.

References.

[Er46b] P. Erdős, On sets of distances of $n$ points, Amer. Math. Monthly 53 (1946), 248–250.
[Er60b] P. Erdős, On sets of distances of $n$ points in Euclidean space, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 165–169.
[Gr56] B. Grünbaum, A proof of Vázsonyi's conjecture, Bull. Res. Council Israel Sect. A 6 (1956), 77–78.
[He56] A. Heppes, Beweis einer Vermutung von A. Vázsonyi, Acta Math. Acad. Sci. Hungar. 7 (1956), 463–466, doi:10.1007/BF02020540.
[HoPa34] H. Hopf and E. Pannwitz, Aufgabe 167, Jber. Deutsch. Math.-Verein. 43 (1934), 114.
[St57] S. Straszewicz, Sur un problème géométrique de P. Erdős, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 39–40.
[Sw09] K. J. Swanepoel, Unit distances and diameters in Euclidean spaces, Discrete Comput. Geom. 41 (2009), 1–27, doi:10.1007/s00454-008-9082-x; arXiv:0707.0213.

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 (21)

Hopf and Pannwitz's planar diameter count Grünbaum's proof of Vázsonyi's conjecture Heppes's proof of Vázsonyi's conjecture Straszewicz's proof of Vázsonyi's conjecture Erdős's asymptotic diameter count in dimension four and above Swanepoel's exact diameter count in dimension four and above problems/distance_problems/E0223/claims Problem 1084 Erdős's asymptotic unit-distance count in dimension four and above Swanepoel's exact unit-distance count in even dimensions six and above Distance Problems distance_problems/erdos_1946_sets_distances_points Theorem 3 (p. 250): among n planar points the diameter occurs at most n times and the minimum distance at most 3n - 6 times distance_problems/erdos_1960_sets_distances_points_euclidean_space Theorem (p. 166): for every k >= 4, g_k(n)/n^2 and G_k(n)/n^2 both tend to 1/2 - 1/(2[k/2])distance_problems/swanepoel_2009_unit_distances_diameters_euclidean_spaces Corollary 3 (p. 4): the exact value of M_d(n) for every d >= 4 and all sufficiently large n Definitions (pp. 1-3): u_d(n), M_d(n), Lenz configurations and extremal sets Theorem 1 (p. 3): for d >= 4 and n >= N(d), extremal sets for unit distances or diameters are Lenz configurations Theorem 4 (p. 4): stability for even d >= 4, near-extremal sets lie mostly on p orthogonal concentric circles Theorem 5 (p. 4): stability for odd d >= 5, near-extremal sets lie mostly on one 2-sphere and p - 1 circles

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