Our main result (Theorem 1) suggests a possible dividing line (μ-superstable + μ-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. #R##N##R##N#Theorem 1. #R##N#Let  KKbe an abstract elementary class with no maximal models of cardinality  μ+μ+which satisfies the joint embedding and amalgamation properties. Suppose  μ≥LS(K)μ≥LS(K). If  KKis μ- and  μ+μ+-superstable and satisfies  μ+μ+-symmetry, then for any increasing sequence  〈Mi∈K≥μ+|i<θ<(sup⁡‖Mi‖)+〉〈Mi∈K≥μ+|i<θ<(sup⁡‖Mi‖)+〉of  μ+μ+-saturated models,  ⋃i<θMi⋃i<θMiis  μ+μ+-saturated.#R##N##R##N##R##N#We also apply results of [18] and use towers to transfer symmetry from μ+μ+ down to μ in abstract elementary classes which are both μ  - and μ+μ+-superstable: #R##N##R##N#Theorem 2. #R##N#Suppose  KKis an abstract elementary class satisfying the amalgamation and joint embedding properties and that  KKis both μ- and  μ+μ+-superstable. If  KKhas symmetry for non-  μ+μ+-splitting, then  KKhas symmetry for non-μ-splitting.
['Monica M. VanDieren']
Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes