Problem 104

Erdős Problem #104: declared status 'open'. Formalized: no. Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$. Current best: Erd\H{o}s claims in a number of places this produces the upper bound $n(n-1)$, but Harborth and Mengerson [HaMe86] note that in fact this delivers an upper bound of $\frac{n(n-1)}{3}$. Prize: $100. OEIS: A003829. Tags: geometry.