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Exercise 5.2.4. in Ziegler, Tent: A Course in Model Theory states:

If (the $L$-structure) $\mathfrak{A}$ is $\kappa$-saturated, then all definable subsets are either finite or have cardinality at least $\kappa$.

This statement seems only correct, if $|\mathfrak{A}|\geq\kappa$, as the set $A$ defined by $\top$ could otherwise have a cardinality $\aleph_0\leq|A|<\kappa$ for $\kappa>\aleph_0$.

I didn't manage to come up with a viable idea for proofing the statement. I would be thankful for any hint.

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A $\kappa$-saturated structure has cardinality at least $\kappa$: otherwise, consider $\{x\not=a: a\in\mathfrak{A}\}$ ... (Fine, unless it's finite.)

So the situation you're envisioning can't occur.

EDIT: And indeed, as Alex observes below, this should be thought of as a hint to the whole problem.

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  • $\begingroup$ Thanks, that clarifed the statement for me. My question - as initially stated - was not clearly formulated: I also was asking for hints proofing the statement. $\endgroup$ – Dalhamun Oct 15 '18 at 18:14
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    $\begingroup$ @Dalhamun You could view Noah's answer as a significant hint to proving the statement in general. The partial type $\{x\neq a : a\in \mathfrak{A}\}$ says that $x$ is not equal to any element of the definable set defined by $\top$... $\endgroup$ – Alex Kruckman Oct 15 '18 at 22:36
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    $\begingroup$ Small correction: a $\kappa$-saturated structure has cardinality at least $\kappa$, unless it is finite. $\endgroup$ – Levon Haykazyan Oct 16 '18 at 14:16
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    $\begingroup$ @Dalhamun What you wrote makes almost no sense to me. Why should $[\varphi(x)]$ consist only of isolated types? $\endgroup$ – Alex Kruckman Oct 16 '18 at 23:54
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    $\begingroup$ @Dalhamun just look at the partial type $\{\varphi(x)\}\cup\{x\neq a \mid a\in \varphi(\mathfrak{A})\}$! Is it realized in $\mathfrak{A}$? $\endgroup$ – Alex Kruckman Oct 16 '18 at 23:56

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