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.
