Sets which do not have subsets of every higher degree

Abstract

Let A be a subset of ω, the set of natural numbers. The degree of A is its degree of recursive unsolvability. We say that A is rich if every degree above that of A is represented by a subset of A. We say that A is poor if no degree strictly above that of A is represented by a subset of A. The existence of infinite poor (and hence nonrich) sets was proved by Soare [9].Theorem 1. Suppose that A is infinite and not rich. Then every hyperarith-metical subset H of ω is recursive in A.In the special case when H is arithmetical, Theorem 1 was proved by Jockusch [4] who employed a degree-theoretic analysis of Ramsey's theorem [3]. In our proof of Theorem 1 we employ a similar, degree-theoretic analysis of a certain generalization of Ramsey's theorem. The generalization of Ramsey's theorem is due to Nash-Williams [6]. If A ⊆ ω we write [A]ω for the set of all infinite subsets of A. If P ⊆ [ω]ω we let H(P) be the set of all infinite sets A such that either [A]ω ⊆ P = ∅. Nash-Williams' theorem is essentially the statement that if P ⊆ [ω]ω is clopen (in the usual, Baire topology on [ω]ω) then H(P) is nonempty. Subsequent, further generalizations of Ramsey's theorem were proved by Galvin and Prikry [1], Silver [8], Mathias [5], and analyzed degree-theoretically by Solovay [10]; those results are not needed for this paper.