On the Bichromatic κ-Set Problem

We study a bichromatic version of the well-known kappa-set problem: given two sets R and B of points of total size it and an integer kappa, how many subsets of the form (R boolean AND h)boolean OR(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 kappa-level in an arrangement of n halfspaces. Disproving an earlier conjecture by Linhart (1993), we present the first nontrivial upper bound for all kappa << n in two dimensions: O(n kappa(1/3) + n(5/6-epsilon) kappa(2/3+2 epsilon) + kappa(2)) for any fixed epsilon > 0. In three dimensions, we obtain the bound O(n kappa(3/2) + n(0.5034)kappa(2.4932) + kappa(3)). Incidentally, this also implies a new tipper bound for the original k-set problem in four dimensions: O(n(2)kappa(3/2) + n(1.5034)kappa(2.4932) + n kappa(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.