Use the <Downward Lowenheim-Skolem theorem> to choose a small elementary submodel $M\preceq N$. By hypothesis, $M$ meets every $E$-class. If there were infinitely many classes, the type
$$
q(x)=\{\neg E(x,m):m\in M\}
$$
would 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 criterion: $E$ has only finitely many classes.

Solution

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

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.


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