On Semi/tame Classes of Finite Structures

We modify the definition of robust chains from [25] as an approach to formalizing the Finite Sub-model Property (see [21], for example) from the perspective of a class of finite structures with a generic model (i.e. having the Amalgamation and Joint-embedding properties). We characterize this formulation, which we call super-robustness, modulo a weaker condition (that is closer to the the situation of a robust chain) that we call robustness. After observing that the more natural condition of a given class is super-robustness up to cofinality – which we call tameness – we prove that a number of natural constructions preserve tameness, and we prove some necessary conditions for tameness refining strong-minimality and total-categoricity. We define a notion of “locally pro-generic” type and verify that this definition provides a re-formulation of þindependence for generic models of certain tame classes. Finally, we show that any class that has the 0,1-law for first-order sentences compatibly with its generic theory is a tame class.