On Stability and Existence of Models in Abstract Elementary Classes

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