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OP-9124OpenGeometry and topology

Erdős #668 · geometry / distances

Is it true that the number of incongruent sets of nn points in R2\mathbb{R}^2 which maximise the number of unit distances tends to infinity as n→∞n\to\infty? Is it always >1>1 for n>3n>3?

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Problem statement

Is it true that the number of incongruent sets of nn points in R2\mathbb{R}^2 which maximise the number of unit distances tends to infinity as n→∞n\to\infty? Is it always >1>1 for n>3n>3?

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Background

In fact this is =1=1 also for n=4n=4, the unique example given by two equilateral triangles joined by an edge.

Computational evidence of Engel, Hammond-Lee, Su, Varga, and Zsámboki [EHSVZ25] and Alexeev, Mixon, and Parshall [AMP25] suggests that this count is =1=1 for various other 5≤n≤215\leq n\leq 21 (although these calculations were checking only up to graph isomorphism, rather than congruency).

The actual maximal number of unit distances is the subject of [90].

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #668 link

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