Π01-classes and Rado's selection principle

There are several areas in recursive algebra and combinatorics in which bounded or recursively bounded -classes have arisen. For our purposes we may define a -class to be a set Path( T ) of all infinite paths through a recursive tree T . Here a recursive tree T is just a recursive subset of ω , the set of all finite sequences of the natural numbers ω = {0,1,2,…}, which is closed under initial segments. If the tree T is finitely branching, then we say the -class Path( T ) is bounded . If T is highly recursive, i.e., if there exists a partial recursive function f : T→ω such that for each node ηЄ T , f (η) equals the number of immediate successors of η, then we say that the -class Path( T ) is recursively bounded (r.b.). For example, Manaster and Rosenstein in [6] studied the effective version of the marriage problem and showed that the set of proper marriages for a recursive society S was always a bounded -class and the set of proper marriages for a highly recursive society was always an r.b. -class. Indeed, Manaster and Rosenstein showed that, in the case of the symmetric marriage problem, any r.b. -class could be represented as the set of symmetric marriages of a highly recursive society S in the sense that given any r.b. Π 1 -class C there is a society S c such that there is a natural, effective, degree-preserving 1:1 correspondence between the elements of C and the symmetric marriages of S c . Jockusch conjectured that the set of marriages of a recursive society can represent any bounded -class and the set of marriages of a highly recursive society can represent any r.b. -class. These conjectures remain open. However, Metakides and Nerode [7] showed that any r.b. -class could be represented by the set of total orderings of a recursive real field and vice versa that the set of total orderings of a recursive real field is always an r.b. -class.