Keisler's Theorem and Cardinal Invariants

We prove that $\operatorname{KT}(\aleph_1)$ implies $\mathfrak{b} = \aleph_1$, where $\operatorname{KT}(\aleph_1)$ is Keisler's theorem for structures of size $\le \aleph_1$. We also establish a connection between hypotheses $\operatorname{KT}(\aleph_0)$ and $\mathrm{SAT}$, which states that suitable ultraproducts are saturated, and hypotheses like $\operatorname{cov}(\mathrm{meager}) = \mathfrak{c}$. Moreover, we prove that $\operatorname{KT}(\aleph_0)$ fails in the random model.