Finding convex sets in convex position

Let F denote a family of pairwise disjoint convex sets in the plane. F is said to be in convex position, if none of its members is contained in the convex hull of the union of the others. For any fixed k greater than or equal to5, we give a linear upper bound on P-k(n), the maximum size of a family F with the property that any k members of F are in convex position, but no n are.