In recent years, methods from (geometric) stability theory have been successfully extended to (possibly unstable) theories without the independence property (NIP theories for short), most prominently in the work of D. Haskell, E. Hrushovski and H. D. Macpherson [Stable domination and independence in algebraically closed valued fields. Cambridge: Cambridge University Press. Lecture Notes in Logic 30 (2008; Zbl 1149.03027)] on the geometric model theory of algebraically closed valued fields. In their study, stably-dominated types play a crucial role. The article under review undertakes a systematic study of a wider class of types, so-called generically stable types, in an arbitrary NIP theory, showing that essential parts of the theory of stably dominated types can be generalised to this wider class of types with a ‘stable-like’ behaviour.
A type \(p\in S(A)\) is called generically stable if there is a non-forking sequence \((b_i)_{i<\omega}\) of realisations of \(p\) (i.e. \(\mathrm{tp}(b_k/Ab_{<k})\) does not fork over \(A\), for all \(k<\omega\)) such that \((b_i)_{i<\omega}\) is an \(A\)-indiscernible set. The main results of the paper concerning this notion are as follows. If \(p\in S(A)\) is generically stable, then it is definable over any infinite non-forking sequence \((b_i)_{i<\omega}\) of realisations of \(p\), the definition scheme being also over \(\mathrm{acl}(A)\). Being generically stable is invariant under parallelism, and generically stable types over algebraically closed sets are stationary, i.e.admit a unique non-forking extension to any superset of the parameters. Non-forking is a symmetric notion when restricted to generically stable types. (In fact, a more general symmetry result is established.)
Apart from these main results, the paper includes a preliminary section on non-forking, non-splitting, definability of types and other basic notions in NIP theories. Furthermore, the author studies the relationship to an earlier approach by S. Shelah [“Classification theory for elementary classes with the dependence property – a modest beginning”, Sci. Math. Jpn. 59, No. 2, 265–316 (2004; Zbl 1081.03031] where a similar notion involving finite satisfiability is used. In fact, the two approaches are shown to coincide for types over models. Many examples and counter-examples are provided throughout the text, illustrating the connections and differences with other ‘stable-like’ notions for types.
In a final section, the author introduces the notion of stable (pre-)weight of a type, as weight with respect to the collection of generically stable types. It is shown that for a type over a model, the stable preweight is bounded by the dependence rank (a rank defined in terms of certain dividing patterns).
We note that there is an overlap of the paper under review with independent work of A. Pillay and E. Hrushovski [“On NIP and invariant measures” (arxiv:0710.2330)].