Shelah's eventual categoricity conjecture in universal classes: Part I

We prove: Theorem 0.1. Let K be a universal class. If K is categorical in cardinals of arbitrarily high cofinality, then K is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (ABCs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions: Theorem 0.2. Let K be an AEC with amalgamation. Assume that K is fully LS(K)-tame and short and has primes over sets of the form M U {a}. Write H-2 := beth(beth)((2)((2LS(K))+)+) . If K is categorical in a lambda > H-2, then K is categorical in all lambda' >= H-2. (c) 2017 Elsevier B.V. All rights reserved.