A Positive Fraction Erdos - Szekeres Theorem

Abstract

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 e 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 e 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). 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p335.pdf yes no no yes