Before revision:  We also show that every set S of n points contains two points such that either (i) every circle passing through them encloses at least points of S, or (ii) every circle passing through them encloses at most \frac{2n-3{3}} \rfloor points of S.
After revision:  We also show that every set S of n points contains two points such that either (i) every circle passing through them encloses at  points of S, or (ii) every circle passing through them encloses at most \frac{2n-3{3}} \rfloor points of S.