Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 3

For $\kappa =|N|$, saturation means that every complete type over a subset of $N$ of cardinality less than $\kappa$ is realized in $N$. The model $N$ is a universal model when every model of $T$ of cardinality less than $\kappa$ elementarily embeds into $N$, and it is a homogeneous model when every partial elementary map between subsets of $N$ of cardinality less than $\kappa$ extends to an automorphism of $N$.

Let $p(x)\in S(A)$ with $A\subseteq N$ and $|A|<\kappa$. Realize $p$ by $c$ in some elementary extension, and use the Downward Lowenheim-Skolem theorem to choose a model $M_0$ containing $A\cup \{c\}$ with $|M_0|<\kappa$. Universality gives an elementary embedding $e:M_0\to N$. Its restriction sends $A$ elementarily to $e(A)$. Homogeneity extends the inverse partial elementary map $e(A)\to A$ to an automorphism $\sigma$ of $N$. Then $\sigma (e(c))$ realizes $p$ over $A$. Thus the universal homogeneous model is saturated.

Use the Downward Lowenheim-Skolem theorem to choose a small elementary submodel $M⪯N$. By hypothesis, $M$ meets every $E$-class. If there were infinitely many classes, the typewould be finitely satisfiable: finitely many parameters meet only finitely many classes, so choose $x$ from another class. Its parameter set is small, so saturation would realize $q$ in $N$. That realization would lie in an $E$-class disjoint from $M$, contradicting the hypothesis. This proves the small elementary submodel meeting every definable equivalence class criterion: $E$ has only finitely many classes.