On the Existence of a Point Subset with 4 or 5 Interior Points

1 Abstract An interior point of a nite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k 1, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing k or k +1 interior points. We proved that h(3) = 3 in an earlier paper. In this paper we prove that h(4) = 7. 2 Introduction Throughout the paper we consider only planar point sets in which no three points are collinear. For such a point set P we distinguish its vertices which lie on the boundary of its convex hull, from the remaining interior points. In 1935, Erd} os and Szekeres 2] proved that for every t 3 there is a number