We review some known results about the Ramsey property 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 of Ramsey's theorem and prove a theorem along these lines.

A standard result about hyperdegrees is proved (under PD) for all ▵ 1 2n + 1 -degrees.

A standard result about hyperdegrees is proved (under PD) for all $\triangle^1_{2n + 1}$-degrees.

A standard result about hyperdegrees is proved for allΔ2n+11-degrees.