Saturated model

A model $N$ is $\kappa$-saturated when every complete type over a subset of size less than $\kappa$ is realized in $N$.

A model is omega-saturated when it realizes every complete finite-tuple type over any finite parameter set. Realizing all one-variable types over finite parameter sets suffices by successively adjoining coordinates. This is weaker than saturation over every countable parameter set.

For every infinite regular cardinal $\kappa$, every first-order structure has a $\kappa$-saturated elementary extension. At one stage use the compactness theorem with the elementary diagram to realize all complete types over subsets of the stage having size less than $\kappa$. There is a set of such types; finite subsets of the enlarged theory are satisfiable because their finitely many witnesses are independent. Iterate for $\kappa$ stages, taking elementary unions at limits. Regularity puts any parameter set of size less than $\kappa$ inside one stage, and its type is realized at the next one. Passing to a larger regular cardinal covers singular requested degrees of saturation as well.

In a countable language, an ultraproduct over a nonprincipal ultrafilter on $\omega$ realizes every countable finitely satisfiable type, including types over countably many parameters. Enumerate the type ${\phi }_1,{\phi }_2,\dots$, and let $A_n$ be the ultrafilter-large set of coordinates where the first $n$ first-order formulas have a simultaneous witness. Put $B_n=\{i:i\ge n\}\cap {\bigcap }_{j\le n}{A}_j$. At coordinate $i$, choose a witness for the largest $n\le i$ with $i\in B_n$, or an arbitrary element when there is none. For each fixed $n$, the resulting function satisfies the first $n$ first-order formulas on $B_n$. The Łoś theorem then realizes the entire type. Countability of the language and parameter set makes the enumeration possible.

A $\kappa$-saturated model has the one-point extension property for partial elementary maps of size below $\kappa$: transport the type of the new element through the map and realize it by saturation. This is $\kappa$-homogeneity; back-and-forth gives the corresponding automorphisms when cardinalities permit.

If an aleph-zero-homogeneous model realizes every finite type over the empty set, it is aleph-zero-saturated. Realize the joint empty-set type of the parameters and a desired element, then use homogeneity to move the realized parameter tuple to the given one.

A model $N$ of cardinality $\kappa$ is universal when every model of the same complete theory having cardinality less than $\kappa$ admits an elementary embedding into $N$.

A model $N$ of cardinality $\kappa$ is homogeneous when every partial elementary map between subsets of size less than $\kappa$ extends to an automorphism of $N$.

A universal homogeneous model realizes every type over a small parameter set. Embed a small model containing a realization into the universal model, then use homogeneity to move the embedded parameters back to the original ones.

Let $N$ be uncountable and saturated, and let a formula define an equivalence relation. If every small elementary submodel of $N$ meets every class, then there are only finitely many classes. Otherwise a small elementary submodel $M$ yields the finitely satisfiable type $\{\neg E(x,m):m\in M\}$, contradicting saturation.