An upper bound on the number of planar K-sets

Abstract

Given a set S of n points, a subset X of size k is called a k-set if there is a hyperplane Π that separates X from S-X. We prove that O(n√k/log*k) is an upper bound for the number of k-sets in the plane, thus improving the previous bound of Erdös, Lovász, Simmons, and Strauss by a factor of log*k. © 1992 Springer-Verlag New York Inc.