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 any fixed k≥ 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, but no n are. In particular, for k=3 , we improve the triply exponential upper bound of T. Bisztriczky and G. Fejes Tóth by showing that P ₃ (n) < 16 ⁿ . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p437.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received March 27, 1997, and in revised form July 10, 1997.