Around stable forking

In the past few years various conjectures have been made concerning the relationship between simple theories and stable theories. The general thrust is that in a simple theory T forking should be accounted for by some kind of “stable fragment” of T . These issues were raised in discussions between Hart, Kim and Pillay in the Fields Institute in the autumn of 1996, but it is quite likely that others have also formulated such problems. The purpose of this paper is to clarify some of these questions and conjectures as well as to prove some relations between them. The theory of local stability, namely the study of φ(x, y)-types where φ(x, y) is a stable formula, will play an important role. The present paper is closely related to the first author’s paper [6], where some positive results are obtained for supersimple theories and simple 1-based theories. One of the properties we will consider is “stable forking”; if a type p(x) ∈ S(M) forks over a subset A of M then this should ∗Supported by NSF grant. †Supported by NSF grant.