Discrete & Computational Geometry

, Volume 19, Issue 3, pp 335-342

First online:

A Positive Fraction Erdos - Szekeres Theorem

  • I. BárányAffiliated withMathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, H-1364 Budapest, Hungary barany@math-inst.hu
  • ,   P. ValtrAffiliated withDepartment of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic valtr@kam.ms.mff.cuni.cz

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We prove a fractional version of the Erdős—Szekeres theorem: for any k there is a constant c k > 0 such that any sufficiently large finite set X⊂ R 2 contains k subsets Y 1 , ... ,Y k , each of size ≥ c k |X| , such that every set {y 1 ,...,y k } with y i ε Y i is in convex position. The main tool is a lemma stating that any finite set X⊂ R d contains ``large'' subsets Y 1 ,...,Y k such that all sets {y 1 ,...,y k } with y i ε Y i 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>