Problem 668

Erdős Problem #668: declared status 'open'. Formalized: no. 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 $n>3$? Current best: Computational evidence of Engel, Hammond-Lee, Su, Varga, and Zs\'{a}mboki [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). Prize: no. OEIS: A385657. Tags: distances, geometry.