 
Summary: Separable Partitions
Noga Alon
Shmuel Onn
Abstract
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 dspace up to a constant factor. Of particular interest are
the values rp,d(n) = (nd(p
2)) for every fixed p and d 3, and rp,2(n) = (n6p12
) for
every fixed p 3. We establish similar results for spaces of finite VapnikChervonenkis
dimension and study the corresponding problem for points on the moment curve as well.
1 Introduction
A separable ppartition of a set of n points in the ddimensional Euclidean space Rd is an
ordered tuple = (1, . . . , p) of p nonempty sets whose disjoint union is S, where the convex
hulls of the sets j are pairwise disjoint. Let rp,d(n) denote the maximum possible number
of separable p partitions of a set of n points in Rd. It is easy to see that rp,1(n) = p! n1
p1 =
(np1). The following theorem determines the asymptotic behavior of rp,d(n) for every
