Non-cupping and randomness
André Nies · Proceedings of the American Mathematical Society · 2006
Let $Y \in \Delta ^0_2$ be Martin-Löf-random. Then there is a promptly simple set $A$ such that for each Martin-Löf-random set $Z$, $Y \le _T A \oplus Z \Rightarrow Y \le _T Z$. When $Y = \Omega$, one obtains a c.e. non-computable set $A$ which is not weakly Martin-Löf cuppable. That is, for any Martin-Löf-random set $Z$, if $\emptyset ’ \le _T A \oplus Z$, then $\emptyset ’ \le _T Z$.