  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 $\mathfrak{c}$.
