On the Number of k-Subsets of a Set of n Points in the Plane

Abstract For a configuration S of n points in E 2 , H. Edelsbrunner (personal communication) has asked for bounds on the maximum number of subsets of size k cut off by a line. By generalizing to a combinatorial problem, we show that for 2 k n the number of such sets of size at most k is at most 2 nk − 2 k 2 − k . By duality, the same bound applies to the number of cells at distance at most k from a base cell in the cell complex determined by an arrangement of n lines in P 2 .