Subsets of superstable structures are weakly benign

Baizhanov and Baldwin [1] introduce the notions of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connect the notion with work of Baldwin, Benedikt, Bouscaren, Casanovas, Poizat, and Ziegler. Stimulated by [1], we investigate here the existence of benign or weakly benign sets. Definition 0.1 1. The set A is benign in M if for every α, β ∈ M if p = tp(α/A) = tp(β/A) then tp * (α/A) = tp * (β/A) where the *-type is the type in the language L * with a new predicate P denoting A. 2. The set A is weakly benign in M if for every α, β ∈ M if p = stp(α/A) = stp(β/A) then tp * (α/A) = tp * (β/A) where the *-type is the type in language with a new predicate P denoting A.