Homogeneous model

A model is strongly $\kappa$-homogeneous when every partial elementary map between its subsets of size less than $\kappa$ extends to an automorphism. Here the unqualified expression homogeneous model uses this automorphism extension convention. For finite parameter tuples it says, in particular, that equal complete types can be carried to one another by an automorphism.

A weaker local convention asks only for one-point extensions of small partial elementary maps. A $\kappa$-saturated model always has that local property. If its cardinality is $\kappa$, the back-and-forth method yields the strong version too. For example, order-preserving maps between finite subsets of the rational dense linear order without endpoints extend to automorphisms by back-and-forth.

A first-order structure has finite-tuple homogeneity when any two finite tuples of the same parameter-free complete type are related by an automorphism. For countably infinite first-order structures this agrees with extension of finite partial elementary maps to automorphisms. Countable atomic models have this property by a back-and-forth method using isolating logical formulas. Uncountable atomic models can fail it.