Some extremal results on circles containing points

Abstract

We define Π(n) to be the largest number such that for every set P of n points in the plane, there exist two points x, y ε P, where every circle containing x and y contains Π(n) points of P. We establish lower and upper bounds for Π(n) and show that [n/27]+2≤Π(n)≤[n/4]+1. We define {Mathematical expression} for the special case where the n points are restricted to be the vertices of a convex polygon. We show that {Mathematical expression}. © 1989 Springer-Verlag New York Inc.