Random non-cupping revisited

Abstract. Say that Y has the strong random anticupping property if there is a set A such that for every Martin-Löf random set R Y ≤T A ⊕ R ⇒ Y ≤T R (in this case A is an anticupping witness for Y). Nies has shown that every random ∆ 0 2 set has the strong random anticupping property via a promptly simple anticupping witness. We show that every ∆ 0 2 set has the random anticupping property via a promptly simple anticupping witness. Moreover, we prove the following stronger statement: for every noncomputable Y ≤T ∅ ′ there exists a promptly simple A such that Y ≤T A ⊕ R ⇒ A ≤T R for all Martin-Löf random sets R. 1