Around stable forking

In the past few years various conjectures have been made concerning therelationship between simple theories and stable theories. The general thrustis that in a simple theory T forking should be accounted for by some kindof \stable fragment" of T. These issues were raised in discussions betweenHart, Kim and Pillay in the Fields Institute in the autumn of 1996, but itis quite likely that others have also formulated such problems. The purposeof this paper is to clarify some of these questions and conjectures as well asto prove some relations between them. The theory of local stability, namelythe study of ˚(x;y)-types where ˚(x;y) is a stable formula, will play animportant role. The present paper is closely related to the rst author’spaper [6], where some positive results are obtained for supersimple theoriesand simple 1-based theories. One of the properties we will consider is \stableforking"; if a type p(x) 2S(M) forks over a subset Aof Mthen this should