A GENERALIZATION OF THE ERDOS-SZEKERES THEOREM TO DISJOINT CONVEX-SETS

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 ally fixed k greate r than or equal to 3, we estimate P-k(n), the maximum size of a family F with the property that any k members of F are in convex position, b ur no n are. In particular, for k = 3, we improve the triply exponenti al upper bound of T. Bisztriczky and G. Fejes Toth by showing that P-3 (n) < 16(n).