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  1. ; ; (, Journal of Mathematical Logic)
    We characterize the completely determined Borel subsets of HYP as exactly the [Formula: see text] subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author. 
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  2. (, Computability)
    We show that if F has a computable instance and the compositional product of F with itself is Weihrauch reducible to F, then F-diamond is Weihrauch reducible to F. The compositional product allows the use of two principles in sequence, while the diamond operator allows an arbitrary but finite number of uses of the given principle in sequence. This answers a question of Pauly. 
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