A Positive Fraction Erdos - Szekeres Theorem

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>