In \cite{Er81d} Erd\H{o}s 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. Erd\H{o}s claims in a number of places this produces the upper bound $n(n-1)$, but Harborth and Mengerson \cite{HaMe86} note that in fact this delivers an upper bound of $\frac{n(n-1)}{3}$.

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

In \cite{Er75h} and \cite{Er92e} Erd\H{o}s also asks how many such unit circles there must be if the points are in general position.

In \cite{Er92e} Erd\H{o}s 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].


References


[El84] Elekes, G., {$n$} points in the plane can determine $n^{3/2}$ unit
circles. Combinatorica (1984), 131.

[Er75h] Erd\H{o}s, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.

[Er81d] Erd\H{o}s, P., Some applications of graph theory and combinatorial methods to number theory and geometry. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.

[Er92e] Erd\H{o}s, P\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.

[HaMe86] Harborth, Heiko and Mengersen, Ingrid, Point sets with many unit circles. Discrete Math. (1986), 193--197.