Let $T$ be a theory in a countable language with a distinguished unary relation
symbol $U$. Given two cardinals $\kappa$ and $\lambda$ we say that $T$ admits the pair
$(\kappa, \lambda)$ if $T$ has a model of size $\kappa$ in which $U$ is interpreted
by a set of size $\lambda$.
Assuming $V=L$ it is known that if $T$ admits the pair $(\kappa^+, \kappa)$ for some
infinite $\kappa$, then it admits this pair for all infinite $\kappa$. This result
is due to Vaught, Chang, and Jensen. In this talk, we explain some aspects of the proof.