Series: Penn State Logic Seminar Date: Tuesday, January 31, 2006 Time: 2:30 - 3:45 PM Place: 106 McAllister Building Speaker: John Clemens, Penn State, Mathematics Title: Weakly Pointed Trees and Partial Injections, part 1 Abstract: We consider some topological and recursion-theoretic questions motivated by the following result of Graf and Mauldin: If X and Y are Polish spaces and B is a Borel subset of X x Y such that for a.e. x the section B_x is uncountable and for a.e. y the section B^y is uncountable, then there is a Borel subset A of X of full measure and a Borel-measurable injection f: A -> Y such that the graph of f is contained in B. We first consider recursion-theoretic results. We introduce a coding of uniformly branching trees and call such a tree T weakly pointed if some branch of the tree can compute T. We show that the set of weakly pointed trees is meager. More precisely, we show that no 2-generic tree can be weakly pointed, but give an example of a 1-generic tree which is weakly pointed. We then consider a topological version of the Graf-Mauldin result, and show that it fails. That is, there is a Borel subset B of X x Y with X and Y Polish spaces such that B_x is uncountable for a comeager set of x and B^y is uncountable for a comeager set of y, but there is no comeager subset A of X and Baire-measurable injection f: A -> Y whose graph is contained in B.