A while ago I asked about versions of Vopenka's principle for logics other than first order. Unfortunately, there doesn't seem to be much there; most logics yield the same notion, while "big" logics like $\mathcal{L}_{\infty\omega}$ yield versions of VP which are easily disprovable.

However, going back I noticed that there is still some potentially interesting stuff around the $\mathcal{L}_{\infty\omega}$ stuff. Say that a class $\mathbb{K}$ of structures has the *weak infinitary VP property* if, whenever $C\subseteq \mathbb{K}$ is a proper class, there are distinct $A, B\in C$ with $A\equiv_{\infty\omega}B$. The *strong infinitary VP property* is the same, except that we ask instead for a $j: A\rightarrow B$ which is an $\mathcal{L}_{\infty\omega}$-elementary embedding, instead of merely $\equiv_{\infty\omega}$. Obviously the strong version is more in the spirit of the original VP; however, the weak version looks interesting too (and more manageable - see below).

Now, it's easy to come up with an example of a class *without* the weak infinitary VP property, via the following fact:

$$\mbox{$A\equiv_{\infty\omega}B$ iff $A\cong B$ in some forcing extension.}$$

Since forcing preserves well-foundedness, no non-isomorphic ordinals can become isomorphic after forcing, so $ON$ does not have the weak infinitary VP property.

However, we can also cook up examples which *do* have it! Let's restrict attention to $\mathbb{K}$ of the form $\{M: M\models T\}$ for some complete first-order theory $T$ in a countable language with no finite models. Then:

Since $\aleph_0$-categoricity is $\Pi^1_2$, it's absolute under forcing. Letting $A, B\in\mathbb{K}$ be of any cardinality, they are therefore isomorphic in any forcing extension where they are both countable - hence $A\equiv_{\infty\omega}B$. So $\mathbb{K}$ has the weak infinitary VP.

Similarly, $\aleph_1$-categorical theories have the weak infinitary VP. The proof is the same, except now the claim that $\aleph_1$-categoricity is absolute is less obvious. On the face of it, "$T$ is $\aleph_1$-categorical" is difficult to express; however, it's equivalent to $T$ being $\omega$-stable and having no Vaughtian pairs, which is absolute.

My question is whether there are any more interesting examples of classes with the weak infinitary VP property, or *any* interesting examples of classes with the strong infinitary VP property. In particular:

Question.Is there an unstable theory with the strong infinitary VP property?

*Note that one thing making the strong infinitary VP property more complicated is that $\mathcal{L}_{\infty\omega}$-elementary embeddability doesn't seem to have as nice a characterization as $\mathcal{L}_{\infty\omega}$-elementary equivalence - at least, none that I know of.*