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.