Keisler's Theorem and Cardinal Invariants

We consider several variants of Keisler's isomorphism theorem. We separate these variants by showing implications between them and cardinal invariants hypotheses. We characterize saturation hypotheses that are stronger than Keisler's theorem with respect to models of size $\aleph_1$ and $\aleph_0$ by $\mathrm{CH}$ and $\operatorname{cov}(\mathsf{meager}) = \mathfrak{c} \land 2^{<\mathfrak{c}} = \mathfrak{c}$ respectively. We prove that Keisler's theorem for models of size $\aleph_1$ and $\aleph_0$ implies $\mathfrak{b} = \aleph_1$ and $\operatorname{cov}(\mathsf{null}) \le \mathfrak{d}$ respectively. As a consequence, Keisler's theorem for models of size $\aleph_0$ fails in the random model. We also show that for Keisler's theorem for models of size $\aleph_1$ to hold it is not necessary that $\operatorname{cov}(\mathsf{meager})$ equals $\mathfrak{c}$.