Shelah’s Eventual Categoricity Conjecture in Tame Aecs with Primes

A new case of Shelah’s eventual categoricity conjecture is established: Theorem 1. LetK be an AEC with amalgamation. WriteH2 := i2i(2LS(K))+ + . Assume that K is H2-tame and K≥H2 has primes over sets of the form M∪{a}. If K is categorical in some λ > H2, then K is categorical in all λ′ ≥ H2. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of Theorem 1 is that Shelah’s categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): Theorem 2. Let D be a homogeneous diagram in a first-order theory T . If D is categorical in a λ > |T |, then D is categorical in all λ′ ≥ min(λ,i(2|T |)+ ).