arXiv:1408.3383v3 [math.LO] 19 Sep 2015 December 2, 2017 TAMENESS, UNIQUENESS TRIPLES AND AMALGAMATION ADI JARDEN Abstract. We combine two approaches to the study of classification theory of AECs: (1) that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and (2) that of Grossberg and VanDieren [8]: (studying non-splitting) as- suming the amalgamation property and tameness. In [9] we derive a good non-forking λ + -frame from a semi-good non- forking λ-frame. But the classes K λ + and ↾ K λ + are replaced: K λ + is restricted to the saturated models and the partial order ↾ K λ + is restricted to the partial order  NF λ + . Here, we avoid the restriction of the partial order ↾ K λ +, assuming that every saturated model (in λ + over λ) is an amalgamation base and (λ, λ + )-tameness for non-forking types over saturated models, (in addition to the hypotheses of [9]): Theorem 7.15 states that M  M + if and only if M  NF λ + M + , provided that M and M + are saturated models. We present sufficient conditions for three good non-forking λ + -frames: one relates to all the models of cardinality λ + and the two others relate to the saturated models only. By an ‘unproven claim’ of Shelah, if we can repeat this procedure ω times, namely, ‘derive’ good non-forking λ +n frame for each n<ω then the categoricity conjecture holds. In [18], Vasey applies Theorem 7.8, proving the categoricity conjec- ture under the above ‘unproven claim’ of Shelah. In [12], we apply Theorem 7.15, proving the existence of primeness triples. Contents 1. Introduction 2 2. Non-forking Frames 5 3. Variants of Tameness 10 4. Deriving Non-Forking Frames Using Tameness 13 5. Continuity Yields Symmetry 16 6. The Relation  NF λ + 18 7. Are the Relations  NF λ + and ↾ K λ + Equivalent? 21 8. Proving Continuity 29 9. Getting Good Non-Forking λ + -Frames 35 10. Meanings of Galois-Types 37 11. Continuations 39 1