  Shelah has provided sufficient conditions for an $L_{\omega_1,
\omega}$-sentence $\psi$ to have arbitrarily large models and for a Morley-like
theorem to hold of $\psi$. These conditions involve structural and
set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney,
Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$
which suffice when $\psi$ is restricted to be universal:
  $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $\psi$ be a
universal $L_{\omega_{1}, \omega}$-sentence.
  - If $\psi$ is categorical in $\aleph_{0}$ and $1 \leq I(\psi, \aleph_{1}) <
2 ^{\aleph_{1}}$, then $\psi$ has arbitrarily large models and categoricity of
$\psi$ in some uncountable cardinal implies categoricity of $\psi$ in all
uncountable cardinals.
  - If $\psi$ is categorical in $\aleph_1$, then $\psi$ is categorical in all
uncountable cardinals.
  The theorem generalizes to the framework of $L_{\omega_1, \omega}$-definable
tame abstract elementary classes with primes.
