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Shore and Slaman (1999) extended this result to all $n \\in \\omega$, by showing that if $S \\nleq_T \\emptyset^{(n-1)}$ then there exists a $G$ such that $S \\oplus G \\geq_T G^{(n)}$. Their argument employs Kumabe-Slaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way. 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