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Suppose I have a model $\mathcal{U}$ with some theory $T$ such that $\mathcal{U}$ is $\kappa$-saturated for some infinite cardinal $\kappa$. If $\mathcal{V} \succeq \mathcal{U}$ is an elementary extension of $\mathcal{U}$, is it true that $\mathcal{V}$ is (at least) $\kappa$-saturated? I.e. is it possible for the saturation to drop when you pass to an elementary extension. One would hope that the saturation doesn't drop but I can't seem to get an argument for why this is the case. Thanks!

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Actually it is not true. $\cal V$ would realise types over subsets of $\cal U$ (since they are realised in $\cal U$) but it may omit other types.

Let $T$ be the theory of dense linear orders without endpoints. Let $\cal U$ be an $\aleph_1$-saturated model of $T$. Consider ${\cal V} = {\cal U} + \mathbb Q$, by which I mean the structure whose universe is the disjoint union of $\mathbb Q$ and the universe of $\cal U$ and every element in $\cal U$ is smaller than any element in $\mathbb Q$. Now $\cal V$ is a dense linear order and hence $\cal U \preceq V$ by model completeness. But $\cal V$ is not $\aleph_1$-saturated, since the type $\{v > p : p \in \mathbb Q\}$ is not realised in $\cal V$.

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