Categoricity from one successor cardinal in Tame Abstract Elementary   Classes
  Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties.   Theorem 1. Suppose K is \chi-tame. If K is categorical in some \lambda^+ >LS(K) then it is categorical in all \mu\geq (\lambda+\chi)^+.   Theorem 2. If K is LS(K)-tame and is categorical both in LS(K) and in LS(K)^+ then K is categorical in all \mu\geq LS(K). 
Rami Grossberg and Monica VanDieren