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,^[1] who observed that tame AECs were much easier to handle than general AECs.

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 ${\displaystyle {\mathfrak {C))}$. Working inside ${\displaystyle {\mathfrak {C))}$, we can define a semantic notion of types by specifying that two elements a and b have the same type over some base model ${\displaystyle M}$ if there is an automorphism of the monster model sending a to b fixing ${\displaystyle M}$ pointwise (note that types can be defined in a similar manner without using a monster model^[2]). 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:

• An AEC ${\displaystyle K}$ is tame if there exists a cardinal ${\displaystyle \kappa }$ such that any two distinct Galois types are already distinct on a submodel of their domain of size ${\displaystyle \leq \kappa }$. When we want to emphasize ${\displaystyle \kappa }$, we say ${\displaystyle K}$ is ${\displaystyle \kappa }$-tame.

Tame AECs are usually also assumed to satisfy amalgamation.

While (without the existence of large cardinals) there are examples of non-tame AECs,^[3] most of the known natural examples are tame.^[4] In addition, the following sufficient conditions for a class to be tame are known:

• Tameness is a large cardinal axiom:^[5] There are class-many almost strongly compact cardinals iff any abstract elementary class is tame.
• Some tameness follows from categoricity:^[6] If an AEC with amalgamation is categorical in a cardinal ${\displaystyle \lambda }$ of high-enough cofinality, then tameness holds for types over saturated models of size less than ${\displaystyle \lambda }$.
• Conjecture 1.5 in ^[7]: If K is categorical in some λ ≥ Hanf(K) then there exists χ < Hanf(K) such that K is χ-tame.

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.^[8] On the other hand, the model theory of tame AECs is much easier to develop, as evidenced by the results presented below.

The following are some important results about tame AECs.

• Upward categoricity transfer:^[9] A ${\displaystyle \kappa }$-tame AEC with amalgamation that is categorical in some successor ${\displaystyle \lambda \geq \operatorname {LS} (K)^{++}+\kappa ^{+))$ (i.e. has exactly one model of size ${\displaystyle \lambda }$ up to isomorphism) is categorical in all ${\displaystyle \mu \geq \lambda }$.
• Upward stability transfer:^[10] A ${\displaystyle \kappa }$-tame AEC with amalgamation that is stable in a cardinal ${\displaystyle \lambda \geq \kappa }$ is stable in ${\displaystyle \lambda ^{+))$ and in every infinite ${\displaystyle \mu }$ such that ${\displaystyle \mu ^{\lambda }=\mu }$.
• Tameness can be seen as a topological separation principle:^[11] An AEC with amalgamation is tame if and only if an appropriate topology on the set of Galois types is Hausdorff.
• Tameness and categoricity imply there is a forking notion:^[12] A ${\displaystyle \kappa }$-tame AEC with amalgamation that is categorical in a cardinal ${\displaystyle \lambda }$ of cofinality greater than or equal to ${\displaystyle \kappa }$ has a good frame: a forking-like notion for types of singletons (in particular, it is stable in all cardinals). This gives rise to a well-behaved notion of dimension.