Problem 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 $n>3$?
Categories:
Geometry Distances
Progress
In fact this is $=1$ also for $n=4$, 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 $=1$ for various other $5\leq n\leq 21$ (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].
Source: erdosproblems.com/668 | Last verified: January 16, 2026