Shelah's Categoricity Conjecture from a successor for Tame Abstract   Elementary Classes
  Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let \mu be the Hanf number of K. Suppose K is tame.   MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^\mu)^+} then K is categorical in all cardinals greater than \beth_{(2^\mu)^+}.   This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]). 
Rami Grossberg and Monica VanDieren