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 the set of homogeneous solutions for a clopen subset $P$ of $[\mathbb{N}]^{\mathbb{N}}$ (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 <i>basis theorem</i>).

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 in identifying the theorem(s) in [3] from which the "basis theorem" follows? 


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[1] <cite authors="Simpson, Stephen G.">_Simpson, Stephen G._, [**Sets which do not have subsets of every higher degree**](http://dx.doi.org/10.2307/2271956), J. Symb. Log. 43, 135-138 (1978). [ZBL0402.03040](https://zbmath.org/?q=an:0402.03040).</cite>

[2] <cite authors="Clote, Peter">_Clote, Peter_, [**A recursion theoretic analysis of the clopen Ramsey theorem**](http://dx.doi.org/10.2307/2274171), J. Symb. Log. 49, 376-400 (1984). [ZBL0574.03030](https://zbmath.org/?q=an:0574.03030).</cite>

[3] <cite authors="Solovay, Robert M.">_Solovay, Robert M._, [**Hyperarithmetically encodable sets**](http://dx.doi.org/10.2307/1997849), Trans. Am. Math. Soc. 239, 99-122 (1978). [ZBL0411.03039](https://zbmath.org/?q=an:0411.03039).</cite>