Sh:300d

• Shelah, S. (2009). Universal Classes: Non-Forking and Prime Modes. In Classification Theory for Abstract Elementary Classes II.
  Ch. V (D) of [Sh:i]
• Abstract:
  We know that failure of smoothness implies non-structure, and also failure of being \chi-based, \chi \ge \text{ LS}({\mathfrak s}). It is not clear how large can be the first \chi for which smoothness fail, and similarly for being \chi-based. But we prove that the failure of at least one of them occurs early. Assuming {\mathfrak s} is smooth and \chi-based for \chi \ge \text{ LS}({\mathfrak s}) we generalize a large part of the theory of the class of |T|^+-saturated models of a stable elementary class, including types of models (not elements), orthogonality and uniqueness of (\mathbb D_{\mathfrak s},\mu)-primary/prime models.
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@inbook{Sh:300d,
 author = {Shelah, Saharon},
 title = {{Universal Classes: Non-Forking and Prime Modes}},
 booktitle = {{Classification Theory for Abstract Elementary Classes II}},
 year = {2009},
 note = {Ch. V (D) of [Sh:i]},
 refers_to_entry = {Ch. V (D) of [Sh:i]}
}