Chains of saturated models in AECs

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough λ: (1) The union of an increasing chain of λ-saturated models is λ-saturated. (2) There exists a type-full good λ-frame with underlying class the saturated models of size λ. (3) There exists a unique limit model of size λ. Our proofs use independence calculus and a generalization of averages to this non first-order context.