Hyperarithmetically encodable sets

Abstract

We say that a set of integers, A, is hyperarithmetically (recursively) encodable, if every infinite set of integers X contains an infinite subset Y in which A is hyperarithmetical (recursive). We show that the recursively encodable sets are precisely the hyperarithmetic sets. Let σ \sigma be the closure ordinal of a universal Σ 1 1 \Sigma _1^1 inductive definition. Then A is hyperarithmetically encodable iff it is constructible before stage σ \sigma . We also prove an effective version of the Galvin-Prikry results that open sets, and more generally Borel sets, are Ramsey, and in the case of open sets prove that our improvement is optimal.