The set of homogeneous solutions of a clopen contains an hyperarithmetical set

In the context of Galvin-Prikry generalization of Ramsey's theorem, I read in a couple of papers ([1],[2]) that Solovay [3] proved that if $$P$$ is a clopen of $$[\mathbb{N}]^{\mathbb{N}}$$ then the set of homogeneous solutions for $$P$$ (i.e. the set of infinite sets $$A$$ s.t. either $$[A]^\mathbb{N}\subset P$$ or $$[A]^\mathbb{N}\cap P = \emptyset$$) always contains a hyperarithmetical set (this is what Clote calls a basis theorem).

I don't see this results being stated explicitly in Solovay's paper, so I guess it follows as a corollary of some other result. Could you please help me in identifying the theorem(s) in [3] from which the "basis theorem" follows?

[1] Simpson, Stephen G., Sets which do not have subsets of every higher degree, J. Symb. Log. 43, 135-138 (1978). ZBL0402.03040.

[2] Clote, Peter, A recursion theoretic analysis of the clopen Ramsey theorem, J. Symb. Log. 49, 376-400 (1984). ZBL0574.03030.

[3] Solovay, Robert M., Hyperarithmetically encodable sets, Trans. Am. Math. Soc. 239, 99-122 (1978). ZBL0411.03039.

• Am I missing something, or is it Theorem 1.8 in [3] you are looking for? – Arno Nov 5 '18 at 21:02
• Maybe...but if there is a solution that lands in $W$ and another one that avoids it I don't see how we could use the theorem 1.8 of [3] to conclude anything on how complicated these solutions are. – Manlio Nov 5 '18 at 21:06