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

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 $\mathrm{tp}(c/A)$ 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.

Let $A=\{a_1,\dots ,a_n\}\subseteq M$ and let $p(x/A)$ be a type. In an elementary extension choose $c$ realizing it and consider the empty-set type $q=\mathrm{tp}(a_1,\dots ,a_n,c)$. By hypothesis, $M$ contains a tuple $(b_1,\dots ,b_n,d)$ realizing $q$. The map $b_i\mapsto a_i$ is partial elementary, so omega-homogeneity extends it to include $d\mapsto d^{\prime }$. Then $d^{\prime }$ realizes $p$. Hence $M$ is omega-saturated, as stated by homogeneity plus realization of empty-set types implies saturation.