A structure $M$ is $\kappa$-saturated when every type over a parameter set of cardinality below $\kappa$ is realized in $M$. It is $\kappa$-homogeneous when every partial elementary map of size below $\kappa$ has the one-point extension property.

Let $f:A\to B$ be such a map and $c\in M$. Transport $\operatorname{tp}(c/A)$ through $f$ to a type over $B$. Its parameter set has size below $\kappa$, so saturation supplies a realization $d$. Then $f\cup\{(c,d)\}$ is partial elementary. This proves that <saturation implies homogeneity>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 144 / 6 / i / Solution

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.


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 Incoming links: Saturation implies homogeneity