Models with few types in infinitary logics

Question

Let $\mathcal{L}_{\kappa \lambda}$ denote the infinitary logic that allows conjunction of less than $\kappa$-many formulas and simultaneous quantification of less than $\lambda$-many variables. It is well-known that

if a sentence $\varphi \in \mathcal{L}_{\omega_1 \omega}$ has arbitrarily large models, then for all cardinals $\lambda \geq \omega$ there exists a model of $\varphi$ of cardinality $\lambda$ which realizes only $\omega$-many types.

(For example, see Theorem 5.11 of Marker's "Lectures in Infinitary Model Theory".)

I would like to have a similar result for $\mathcal{L}_{\kappa^+ \kappa}$ with uncountable $\kappa$ that guarantees the existence of sufficiently large models with $\leq \kappa$-many types. (Indeed, having $\leq \kappa$-many quantifier free 1-types suffices for my purposes.)

The problem I am having is that the usual proof of this fact does not generalize to uncountable $\kappa$: The proof proceeds by taking the Skolem hull of a large set of indiscernibles, whose existence is guaranteed by an argument that uses the Erdös-Rado theorem. If one tries to imitate the same proof, in order to guarantee that the Skolem hull is an elementary substructure, one has to introduce Skolem functions of infinite arity, which in turn requires one to consider indiscernibles with respect to formulas possibly having infinitely many variables; and this requires a partition relation where one colors $\geq \omega$-tuples. However, such partition relations are known to fail in general.

Is there any way to get around this problem or is it the case that the generalization of this fact fails? If it helps, you may assume that $\kappa$ has various properties such as being strong limit, inaccessible etc.

Answer 1

Here is a partial answer: consistently, the generalization can fail for all uncountable $\kappa$. Namely:

Suppose $\mathbb{V} = \mathbb{L}$ and let $\kappa$ be any uncountable cardinal. Let $\mathcal{L}$ be the language consisting of the binary relation $\epsilon$ and constant symbols $(c_\alpha: \alpha < \kappa)$. Let $\phi$ be a sentence of $\mathcal{L}_{\kappa^+ \omega_1}$ such that $(M, \epsilon, a_\alpha: \alpha < \kappa) \models \phi$ iff $(M, \epsilon)$ is a well-founded model of $ZFC^- + \mathbb{V}=\mathbb{L}$, and $a_\alpha$ is the $\alpha$'th element of $\mbox{ON}^M$ (here $ZFC^-$ is $ZFC$ without powerset, or any other strong enough fragment of $ZFC$ for condensation to work).

For each ordinal $\gamma \geq \kappa$ let $M_\gamma$ be the $\mathcal{L}$-structure with universe $\mathbb{L}_\gamma$, where $\epsilon$ is interpreted as $\in$ and where $c_\alpha$ is interpreted as $\alpha$. Then every model of $\phi$ is isomorphic to some $M_\gamma$. But if $|M_\gamma| \geq \kappa^+$ then $\mathcal{P}(\kappa) \subseteq M_\gamma$, and so $M_\gamma$ realizes $2^\kappa$ distinct quantifier-free $1$-types.