In [Er81d] Erdős proved that $\gg n$ many circles is possible, and that there cannot be more than $O(n^2)$ many circles. The argument is very simple: every pair of points determines at most $2$ unit circles, and the claimed bound follows from double counting. Erdős 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 $\frac{n(n-1)}{3}$.

Elekes [El84] has a simple construction of a set with $\gg n^{3/2}$ such circles. This may be the correct order of magnitude.

In [Er75h] and [Er92e] Erdős also asks how many such unit circles there must be if the points are in general position.

In [Er92e] Erdős offered £100 for a proof or disproof that the answer is $O(n^{3/2})$.

The maximal number of unit circles achieved by $n$ points is A003829 in the OEIS.

See also [506] and [831].

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