Lines with Many Points on Both Sides

Abstract

Let G be a finite set of points in the plane. A line M is a (k, k)-line if M is determined by G, and there are at least k points of G in each of the two open half-planes bounded by M. Let f(k, k) denote the maximum size of a set G in the plane, which is not contained in a line and does not determine a (k, k)-line. In this paper we improve previous results of Yaakov Kupitz (f(k, k) ≤ 3k), Noga Alon f(k, k) ≤ 2k + O(√k)), and Micha A. Perles (f(k, k)≤2k + O(log k)). We show that f(k,k)≤ 2k+O(log log k).