On unit circles which avoid all but two points of a given point-set (Q1348763)

The paper deals with an analogue of the following Gallai theorem of 1944, which solves the well-known Sylvester problem of 1893: For a finite set of at least two non-collinear points in a plane, there exists a line passing through exactly two of the points. The analogue concerns circles: Theorem. For any finite set of at least two points in the plane which has diameter less than \(\sqrt 3\), there is a unit circle passing through exactly two points of the set.