Separable Partitions

An ordered partition of a set of n points in the d dimensional Euclidean space is called a separable partition if the convex hulls of the parts are pairwise disjoint. For each fixed p and d we determine the maximum possible number rp,d(n) of separable partitions into p parts of n points in real d-space up to a constant factor. Of particular interest are the values rp,d(n) = Θ(nd( p 2)) for every fixed p and d ≥ 3, and rp,2(n) = Θ(n6p−12) for every fixed p ≥ 3. We establish similar results for spaces of finite Vapnik-Chervonenkis dimension and study the corresponding problem for points on the moment curve as well.