Problem 217

Erdős Problem #217: declared status 'open'. Formalized: no. For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times? Current best: Erd\H{o}s originally believed this was impossible for $n\geq 5$, but Pomerance constructed a set with $n=5$ (see [Er83c] for a description), and Pal\'{a}sti has proved such sets exist for all $n\leq 8$. Prize: no. Tags: distances, geometry.