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Problem 668 — Erdős Problems

Problem 668

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 $n>3$ ?

Source: erdosproblems.com/668

Geometry
Distances

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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).

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Status: Open
Topics: Geometry
Distances
In the wiki: Wiki page