A Positive Fraction Erdos - Szekeres Theorem

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image). We prove a fractional version of the Erds-Szekeres theorem: for any k there is a constant c sub(k) > 0 such that any sufficiently large finite set X R super(2) contains k subsets Y sub(1) , ... ,Y sub(k) , each of size greater than or equal to c sub(k) |X| , such that every set {y sub(1) ,...,y sub(k) } with y sub(i) epsilon Y sub(i) is in convex position. The main tool is a lemma stating that any finite set X R super(d) contains "large" subsets Y sub(1) ,...,Y sub(k) such that all sets {y sub(1) ,...,y sub(k) } with y sub(i) epsilon Y sub(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).