= Universal homogeneous model is saturated
Let $T$ be a <complete theory>, let $\kappa\geq|T|+\aleph_0$, and let $N\models T$ have cardinality $\kappa$. If $N$ is a <universal model> for all models of $T$ of size at most $\kappa$, and every <partial elementary map> of size less than $\kappa$ extends to an <automorphism>, then $N$ is $\kappa$-<saturated model>[saturated].
Let $p(x)$ be a <complete type> over $A\subseteq N$, with $|A|<\kappa$. By <compactness theorem> it is realized by an element $b$ of an <elementary extension> of $N$. By the <Downward Lowenheim-Skolem theorem>, there is an elementary substructure $M$ containing $A\cup\{b\}$ with $|M|\leq\kappa$. Universality gives an <elementary embedding> $f:M\to N$. The inverse map from $f(A)$ to $A$ is partial elementary, so an <automorphism> $\sigma$ of $N$ extends it. Then $\sigma(f(b))$ realizes $p$ over the original set $A$.
Using only universality for models strictly smaller than $\kappa$ would leave a gap when the language itself has size $\kappa$, and can be vacuous for countable models.
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