When does ${\aleph }_1$-categoricity imply $\omega$-stability?

Abstract

For an ${\aleph }_1$-categorical atomic class, we clarify the space of types over the unique model of size ${\aleph }_1$. Using these results, we prove that if such a class has a model of size ${\beth }_1^{+}$ then it is $\omega$-stable.

22 pages