F-sigma subset of plane meeting every circle at 3 points

Question

Is there an $F_{\sigma}$-set (countable union of closed subsets of plane) $S \subseteq \mathbb{R}^2$ that meets every circle at 3 points?

Answer 1

There is not. If there were an $F_\sigma$ set meeting every circle in three points, then a suitable Moebius transformation would turn it into an $F_\sigma$ set meeting every line in three points. Bouhjar, Dijkstra, and Maudlin proved that no set meeting every line in exactly three points can be $F_\sigma$, extending an old result of Larman that says the same thing but for sets meeting every line in two points.