Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes

This dissertation examines three main topics, the topic of defining "superstability" for abstract elementary classes (AECs), uniqueness of limit models, and two cardinal models in abstract elementary classes. In particular we further generalize an analogue of Vaught's theorem which constructs an uncountable two cardinal model starting from the existence of a countable Vaughtian pair in an elementary class to the AEC context originally published by Lessmann, who in turn built upon the work of Grossberg and VanDieren, Shelah, and others. We also give various sufficient conditions on countable models, as well as a condition on models of size kappa that, assuming that a simplified morass,ñ allows us to construct a (kappa^++,kappa)-model. We discuss how this work in AECs to some degree parallels the proof of Jensen's Gap-2 transfer theorem for elementary classes. We also discuss difficulties inherent in proving a true gap-2 transfer theorem for AECs. Additionally, we discuss, progress that has been made toward proving the uniqueness of limit models assuming various "superstability-like" assumptions (much of the work described is due to Shelah, Villaveces, Grossberg, and VanDieren). One small original result is contributed to this discussion.