The Ramsey property and degrees in the analytical hierarchy

In Chapter I we review some known results about the Ramsey theory for partitions of reals, and we present a certain two-person game such that if either player has a winning strategy then a homogeneous set for the partition can be constructed, and conversely. This gives alternative proofs of some of the known results. We then discuss possible uses of the game in obtaining effective versions and prove a theorem along these lines. In Chapter II we study the structure of initial segments of the Δ12n+1-degrees, assuming Projective Determinacy. We show that every finite distributive lattice is isomorphic to such an initial segment, and hence that the first-order theory of the ordering of Δ12n+1-degrees is undecidable. In Chapter III we extend Friedberg's Jump Inversion theorem to Q2n+1-degrees, after noticing that it fails tor Δ12n+1-degrees. We assume again Projective Determinacy.