On Stability and Existence of Models in Abstract Elementary Classes

Abstract: For an abstract elementary class $\mathbf{K}$ and a cardinal $\lambda \geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $\lambda^+$-minimal types and continuity of splitting in $\lambda$, that stability in $\lambda$ is equivalent to the existence of a model in $\lambda^{++}$. The forward direction holds without any cardinal or categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24, 3.14]. Moreover, we prove a categoricity theorem for abstract elementary classes with weak amalgamation and tameness under mild structural assumptions in $\lambda$. A key feature of this result is that we do not assume amalgamation or arbitrarily large models.