Hart, B.; Pillay, Anand; Starchenko, S. \(1\)-based theories – the main gap for \(a\)-models. (English) Zbl 0849.03022 Arch. Math. Logic 34, No. 5, 285-300 (1995). For a 1-based theory \(T\) with NDOP, any \(a\)-\(k_p(T)\)-saturated model is proved to be prime (in the category of \(a\)-\(k_p(T)\)-saturated models) over a non-forking tree of elements, where the tree height is at most \(|T|^+\) (usually called a structure theorem). For a shallow theory of this kind, the number of \(a\)-\(k_p(T)\)-saturated models of \(T\) of cardinality \(\aleph_a\) is less than \({\mathcal J}_{(2^{|T|})^+} (|\omega + \alpha|)\). Otherwise \(T\) has \(2^\lambda\) of \(a\)-\(k_p(T)\)-saturated models of cardinality \(\lambda\), for most \(\lambda\). Reviewer: A.Ryaskin (Novosibirsk) Cited in 1 Document MSC: 03C45 Classification theory, stability, and related concepts in model theory 03C50 Models with special properties (saturated, rigid, etc.) Keywords:forking; deep theory; orthogonality; prime model; saturated model; regular type; 1-based theory; NDOP; shallow theory PDFBibTeX XMLCite \textit{B. Hart} et al., Arch. Math. Logic 34, No. 5, 285--300 (1995; Zbl 0849.03022)