FOM: question

[marker@math.uic.edu]

In his November 3, message Harvey states one of my favorite theorems on f.o.m.

A) Let E be a Borel subset of the unit square in the plane, which is symmetric
about the line y = x. Then E contains or is disjoint from the graph of a
Borel measurable function.

Harvey showed that A) is true but not provable in (or well beyond) second order arithmetic.

For those of you who don't know the proof that A) is true it is an easy consequence of the  
determinacy of Borel games.  This made me wonder if anyone of you knows the answers
to the following two questions that came to mind.

1) Is A) equivalent to Borel determinacy over second order arithmetic (or some natural
fragment)?

B) Let E be an analytic subset of the unit square in the plane, which is symmetric
about the line y = x. Then E contains or is disjoint from the graph of a
Borel measurable function.  (Unlike my previous post, here by "analytic" I mean the  
projection of a Borel set in R^3.)

B) is again an easy consequence of analytic determinacy. By theorems of Martin and  
Harrington (another f.o.m. highlight) we know that analytic determinacy is equivalent to the  
existence of sharps.

2) Is B) equivalent to analytic determinacy? (over ZFC, second order artithemtic...)

Dave Marker