On the bichromatic k -set problem

Abstract

We study a bichromatic version of the well-known k-set problem : given two sets R and B of points of total size n and an integer k , how many subsets of the form (R ∩ h ) ∪ ( B ∖ h ) can have size exactly k over all halfspaces h ? In the dual, the problem is asymptotically equivalent to determining the worst-case combinatorial complexity of the k-level in an arrangement of n halfspaces . Disproving an earlier conjecture by Linhart [1993], we present the first nontrivial upper bound for all k ≪ n in two dimensions: O ( nk 1/3 + n 5/6−ϵ k 2/3+2 ϵ + k 2 ) for any fixed ϵ<0. In three dimensions, we obtain the bound O ( nk 3/2 + n 0.5034 k 2.4932 + k 3 ). Incidentally, this also implies a new upper bound for the original k -set problem in four dimensions: O ( n 2 k 3/2 + n 1.5034 k 2.4932 + n k 3 ), which improves the best previous result for all k ≪ n 0.923 . Extensions to other cases, such as arrangements of disks, are also discussed.