When does $\aleph_1$-categoricity imply $\omega$-stability?

TL;DR

This paper investigates the relationship between $eth_1^+$-sized models and $ ext{omega}$-stability in $oldsymbol{eth_1}$-categorical atomic classes, providing conditions under which categoricity implies stability.

Contribution

It establishes that $eth_1^+$-sized models in $eth_1$-categorical atomic classes ensure $ ext{omega}$-stability, clarifying the connection between model size and stability.

Findings

If an $eth_1$-categorical atomic class has a model of size $eth_1^+$, then it is $ ext{omega}$-stable.

The paper characterizes the space of types over the unique $eth_1$-sized model.

Provides new insights into the structure of models in categorical classes.

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.