Does there exist a $\kappa$-compact elementary extension of cardinality $\kappa$ of a given structure?

Question

It is well-known that there does necessarily not exist a $\kappa$-saturated elementary extension of cardinality $\kappa$ of a given structure (if, for instance, if the structure is unstable.)

What if we weaken the requirement from saturation to compactness? Here, $M$ being $\kappa$-compact means that it realizes all partial type of size less than $\kappa$.

It seems that there still seem to be too many partial types of the required size, so there may not exist such a thing. Is there a reference on this matter?

Answer 1

You note correctly that $\kappa$-compact is a weaker notion than $\kappa$-saturated (for infinite $\kappa$).

Proof: Suppose $\Sigma$ is a partial type with $|\Sigma|<\kappa$. Then if $A$ is the set of parameters mentioned by $\Sigma$, we have $|A|<\kappa$. So we can extend $\Sigma$ to a complete type $p$ over $A$, and any model realizing $p$ realizes $\Sigma$. So any $\kappa$-saturated model is $\kappa$-compact. $\square$

The converse is true when $\kappa>|L|$ (by $|L|$, I mean the number of formulas with no parameters).

Proof: Suppose $p$ is a complete type over $A$, with $|A|<\kappa$. Then $|p| = \max(|L|,|A|)<\kappa$, and any $\kappa$-compact model realizes $p$. So any $\kappa$-compact model is $\kappa$-saturated. $\square$

So if $\kappa>|L|$ is such that $M$ has no $\kappa$-saturated elementary extension of size $\kappa$, then $M$ also has no $\kappa$-compact elementary extension of size $\kappa$!

OK, what about when $\kappa \leq |L|$?

1. When $|L| = \aleph_0$, the situation is trivial, since every structure is $\aleph_0$-compact: a finite partial type is equivalent to a formula, and every consistent formula is realized in every model. So for countable languages, $\kappa$-saturation is clearly the right notion: it's equivalent to $\kappa$-compactness for uncountable $\kappa$, and $\aleph_0$-saturated is meaningful, while $\aleph_0$-compactness is not.
2. Now suppose $|L|$ is uncountable. When $\kappa < |L|$, you're asking about models which are smaller than the size of the language, and it's usually pretty hopeless to say anything general about these models (basically because we can't use Löwenheim-Skolem).
3. So the remaining question is about $|L|$-compact models, for uncountable $|L|$. Let $\kappa = |L|$. Theorem 1.7(1) in Classification Theory tells us that if $\kappa^{<\kappa} = \kappa$, then for any infinite model $M$ with $|M|\leq \kappa$, $M$ has a $\kappa$-compact elementary extension of size $\kappa$ (the proof is just iteratively realizing partial types). Also, the same is true if $T$ is $\kappa$-stable (we can even get a saturated extension in this case). I'm not sure about the converse... Certainly there are $\kappa^{<\kappa}$ partial types over $M$, but even if this is greater than $\kappa$, it's plausible that we can realize them all by only adding $\kappa$-many elements, because they're not necessarily pairwise inconsistent. You can interpret this as a topological question about $S(M)$. Is there a set $X\subseteq S(M)$ of size $\leq \kappa$ with the following weird density property: Every closed set formed as an intersection of $<\kappa$ basic clopen sets has nonempty intersection with $X$? I couldn't find a place in Classification Theory where this question is addressed.