A positive fraction Erdos-Szekeres theorem

Abstract

We prove a fractional version of the Erdos-Szekeres theorem: for any k there is a constant ck > 0 such that any sufficiently large finite set X ⊂ ℝ2 contains k subsets Yl,...,Yk, each of size ≥ ck |X|, such that every set {yl,...,yk} with yi ∈ Yi is in convex position. The main tool is a lemma stating that any finite set X ⊂ ℝd contains "large" subsets Yl,...,Yk such that all sets {yl,...,yk} with yi ∈ 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).