THE ERDOS-SZEKERES PROBLEM FOR NON-CROSSING CONVEX SETS

Abstract

We show an equivalence between a conjecture of Bisztriczky and Fejes T ' oth about families of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and T ' oth on the Erdos-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdos-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erdos-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd " os-Szekeres theorem of P or and Valtr to families of non-crossing convex bodies.