A positive fraction Erdos-Szekeres theorem

We prove a fractional version of the Erdos-Szekeres theorem: for any k there is a constant c(k) > 0 such that any sufficiently large finite set X subset of R-2 contains k subsets Y-1, ..., Y-k, each of size greater than or equal to c(k)\X\, such that every set {y(1), ..., y(k)} with y(i) is an element of Y-i is in convex position. The main tool is a lemma stating that any finite set X subset of R-d contains "large" subsets Y-1, ..., Y-k such that all sets {y(1), ..., y(k)} with y(i) is an element of 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).