A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets

  title={A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets},
  author={J{\'a}nos Pach and G{\'e}za T{\'o}th},
  journal={Discrete & Computational Geometry},
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 xed 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 TT oth by showing that P 3 (n) < 16 n . 

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