Problem 668

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claims/: The 1 claim page of Problem 668, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\to\infty$? Is it always $>1$ for $n>3$?

Status. Open.

Source. erdosproblems.com/668, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #668, https://www.erdosproblems.com/668.

References.

• [AMP25] B. Alexeev, D. Mixon, and H. Parshall, The Erdős unit distance problem for small point sets. arXiv:2412.11914 (2025).
• [EHSVZ25] P. Engel, O. Hammond-Lee, Y. Su, D. Varga, and P. Zsámboki, Diverse beam search to find densest-known planar unit distance graphs. arXiv:2406.15317 (2025).

Formalization. None recorded.

Current assessment

• First question. Whether the number of incongruent maximizers tends to infinity is open.
• Second question. It fails at $n=4$: five unit distances among four points occur only for the rhombus of two unit equilateral triangles, as the claim page for the site's remark records. That claim is pending, so the standing stays open, with this part settled by a pending claim.
• Small cases. [AMP25] (Theorem 1(c), Table 2) lists every densest unit-distance graph on at most $21$ vertices up to isomorphism. It finds one graph for $n=5,7,9,10,12,13,15,16,20$; for those $n$ the count of incongruent maximizers is undetermined, since one graph may have incongruent realizations. It finds several graphs for $n=6,8,11,14,17,18,19,21$, so the count exceeds one for those $n$. As the site notes, [EHSVZ25] and [AMP25] count graphs up to isomorphism, not point sets up to congruence.

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.

• alexeev_2024_erdos_unit_distance_problem_small_point