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Updated Oct 8

Problem 605

claims/: The 2 claim pages of Problem 605, one per claimant's result; the problem's standing derives from them.

Statement. Is there some function $f(n)\to \infty$ as $n\to\infty$ such that there exist $n$ distinct points on the surface of a two-dimensional sphere with at least $f(n)n$ many pairs of points whose distances are the same?

Status. Proved.

Source. erdosproblems.com/605, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #605, https://www.erdosproblems.com/605.

References.

[EHP89] Erdős, Paul and Hickerson, Dean and Pach, János, A problem of Leo Moser about repeated distances on the sphere. Amer. Math. Monthly (1989), 569-575.
[SwVa04] Swanepoel, Konrad J. and Valtr, Pavel, The unit distance problem on spheres. (2004), 273-279.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

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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 (8)

Erdős, Hickerson and Pach's superlinear repeated distance on the sphere Swanepoel and Valtr's root-log lower bound on spheres problems/distance_problems/E0605/claims Distance Problems distance_problems/erdos_1989_problem_leo_moser_about_repeated_distances Theorem 2 (p. 572): n points on S^2 with each at distance α from c_1 log* n others, and at distance √2 from c_2 n^{1/3} others distance_problems/swanepoel_2004_unit_distance_problem_spheres Theorem 1 (p. 1): u_D(n) > cn sqrt(log n) unit distances on every sphere of diameter D > 1

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