Title: Vaught's Two-Cardinal Theorem and Quasi-Minimality in Continuous Logic,
Abstract: We prove the following continuous analogue of Vaught's Two-Cardinal Theorem:
if for some $\kappa>\lambda\geq \aleph_0$, a continuous theory $T$ has a model
with density character $\kappa$ which has a definable subset of density
character $\lambda$, then $T$ has a model with density character $\aleph_1$
which has a separable definable subset. We also show that if we assume that $T$
is $\omega$-stable, then if $T$ has a model of density character $\aleph_1$
with a separable definable set, then for any uncountable $\kappa$ we can find a
model of $T$ with density character $\kappa$ which has a separable definable
subset. In order to prove this, we develop an approximate notion of
quasi-minimality for the continuous setting. We apply these results to show a
continuous version of the forward direction of the Baldwin-Lachlan
characterization of uncountable categoricity: if a continuous theory $T$ is
uncountably categorical, then $T$ is $\omega$-stable and has no Vaughtian
pairs.