Universal classes near ${\mathrm{\aleph }}_1$

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 ${\mathrm{\aleph }}_n$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\mathrm{\aleph }}_0$ and ${\mathrm{\aleph }}_1$ which suffice when $\psi$ is restricted to be universal: $\mathbf{T}\mathbf{h}\mathbf{e}\mathbf{o}\mathbf{r}\mathbf{e}\mathbf{m}$ Assume $2^{{\mathrm{\aleph }}_0}<2^{{\mathrm{\aleph }}_1}$. Let $\psi$ be a universal $L_{{\omega }_1,\omega }$-sentence.

• If $\psi$ is categorical in ${\mathrm{\aleph }}_0$ and $1 \leq I(\psi, \aleph_{1}) < 2 ^{\aleph_{1}}$,then$\psi$hasarbitrarilylargemodelsandcategoricityof$\psi$insomeuncountablecardinalimpliescategoricityof$\psi$ in all uncountable cardinals.
• If $\psi$ is categorical in ${\mathrm{\aleph }}_1$, 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.