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Abstract: A new case of Shelah's eventual categoricity conjecture is established: Let K be an abstract elementary class with amalgamation. Write μ : = ℶ ( 2 LS ( K ) ) + and H 2 : = ℶ ( 2 μ ) +. Assume that K is H2‐tame and K ≥ H 2 has primes over sets of the form M ∪ { a }. If K is categorical in some λ > H 2, then K is categorical in all λ ′ ≥ H 2. 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 the mentioned result is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): If D be a homogeneous diagram in a first‐order theory T and D is categorical in a λ > | T |, then D is categorical in all λ ′ ≥ min ( λ , ℶ ( 2 | T | ) + ). [ABSTRACT FROM AUTHOR]

Abstract: In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for μ‐forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking‐like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent. [ABSTRACT FROM AUTHOR]

We show how to build prime models in classes of saturated models of abstract elementary classes (AECs) having a well-behaved independence relation: Let [ABSTRACT FROM AUTHOR]