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THE ERDŐS–SZEKERES PROBLEM FOR NON-CROSSING CONVEX SETS

Part of: Discrete geometry

Published online by Cambridge University Press:  14 May 2014

Michael Gene Dobbins ,
Andreas Holmsen  and
Alfredo Hubard
Michael Gene Dobbins
Affiliation:
GAIA, Postech, Pohang, South Korea email dobbins@postech.ac.kr
Andreas Holmsen
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon, South Korea email andreash@kaist.edu
Alfredo Hubard
Affiliation:
Département d’informatique, École Normale Supérior, Paris, France email hubard@di.ens.fr

Abstract

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We show an equivalence between a conjecture of Bisztriczky and Fejes Tóth 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 Tóth on the Erdős–Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdős–Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erdős–Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erdős–Szekeres theorem of Pór and Valtr to families of non-crossing convex bodies.

MSC classification

Secondary: 52C10: Erdős problems and related topics of discrete geometry 52C30: Planar arrangements of lines and pseudolines

Information

Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 463 - 484
Copyright
Copyright © University College London 2014 

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