On Stability and Existence of Models in Abstract Elementary Classes

For an abstract elementary class K and a cardinal λ ≥ LS(K), we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for λ⁺-minimal types and continuity of splitting in λ, that stability in λ is equivalent to the existence of a model in λ⁺⁺. 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 λ. A key feature of this result is that we do not assume amalgamation or arbitrarily large models.