Uniqueness of Limit Models in Classes with Amalgamation
  Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the L\"owenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, for the precise description of long splitting chains and locality}, then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M.   This theorem extends results of Shelah, Kolman and Shelah, and Shelah and Villaveces. A preliminary version of our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes. 
R. Grossberg (1), M. VanDieren (2), A. Villaveces (3) ((1) Carnegie   Mellon University, (2) Robert Morris University, (3) U...