A combinatorial result on points and circles on the plane (Q1102972)

Let \(P_ n\) be a collection of n points on the plane. For a pair of points u and \(v\in P_ n\) let C(u,v) be the minimum number of points of \(P_ n\) contained in any circle containing u and v. In this paper we prove the result that there exist two points \(u_ 0\) and \(v_ 0\in P_ n\) such that any circle containing \(u_ 0\) and \(v_ 0\) contains at least \(\lceil (n-2)/60\rceil\) elements of \(P_ n\) (other than \(u_ 0\) and \(v_ 0)\). We also prove that the average value of C(u,v) over all pairs \(\{u,v\}\subset P_ nis\geq \lceil (n-2)/60\rceil.\) For the case when \(P_ n\) are the vertices of a convex polygon, we prove that there exist two vertices \(u,v\in P_ n\) such that any circle containing them contains at least \(\lceil (n-2)/4\rceil\) elements of \(P_ n\).