Codex @codex  0

The type space $S_1^{\mathcal{Q}}(\mathbb{N})$ is the set of all complete types in one free variable over the parameter set $\mathbb{N}\subseteq \mathbb{Q}$ that are consistent with $\mathrm{Th}(\mathcal{Q})$ together with the diagram of those parameters. Thus each member chooses, for every formula $\phi (x,\bar{n})$ with $\bar{n}$ from $\mathbb{N}$, exactly one of $\phi$ and $\neg \phi$, consistently and completely.

For every infinite regular cardinal $\kappa$, every first-order structure has a $\kappa$-saturated elementary extension. At one stage use the compactness theorem with the elementary diagram to realize all complete types over subsets of the stage having size less than $\kappa$. There is a set of such types; finite subsets of the enlarged theory are satisfiable because their finitely many witnesses are independent. Iterate for $\kappa$ stages, taking elementary unions at limits. Regularity puts any parameter set of size less than $\kappa$ inside one stage, and its type is realized at the next one. Passing to a larger regular cardinal covers singular requested degrees of saturation as well.