Tame abstract elementary class

In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren, who observed that tame AECs were much easier to handle than general AECs.
Definition.
Let "K" be an AEC with joint embedding, amalgamation, and no maximal models. Just like in first-order model theory, this implies "K" has a universal model-homogeneous monster model formula_1. Working inside formula_1, we can define a semantic notion of types by specifying that two elements "a" and "b" have the same type over some base model formula_3 if there is an automorphism of the monster model sending "a" to "b" fixing formula_3 pointwise (note that types can be defined in a similar manner without using a monster model). Such types are called Galois types.
One can ask for such types to be determined by their restriction on a small domain. This gives rise to the notion of tameness:
Tame AECs are usually also assumed to satisfy amalgamation.
Discussion and motivation.
While (without the existence of large cardinals) there are examples of non-tame AECs, most of the known natural examples are tame. In addition, the following sufficient conditions for a class to be tame are known:
Many results in the model theory of (general) AECs assume weak forms of the Generalized continuum hypothesis and rely on sophisticated combinatorial set-theoretic arguments. On the other hand, the model theory of tame AECs is much easier to develop, as evidenced by the results presented below.
Results.
The following are some important results about tame AECs.