Discrete & Computational Geometry

, Volume 19, Issue 3, pp 335–342

A Positive Fraction Erdos - Szekeres Theorem

  • I. Bárány
  •   P. Valtr

DOI: 10.1007/PL00009350

Cite this article as:
Bárány, I. & P. Valtr Discrete Comput Geom (1998) 19: 335. doi:10.1007/PL00009350


We prove a fractional version of the Erdős—Szekeres theorem: for any k there is a constant ck > 0 such that any sufficiently large finite set X⊂R2 contains k subsets Y1, ... ,Yk , each of size ≥ ck|X| , such that every set {y1,...,yk} with yiε Yi is in convex position. The main tool is a lemma stating that any finite set X⊂Rd contains ``large'' subsets Y1,...,Yk such that all sets {y1,...,yk} with yiε Yi have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem). <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>19n3p335.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>

Copyright information

© 1998 Springer-Verlag New York Inc.

Authors and Affiliations

  • I. Bárány
    • 1
  •   P. Valtr
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, H-1364 Budapest, Hungary barany@math-inst.huHU
  2. 2.Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic valtr@kam.ms.mff.cuni.czCZ