Tameness, uniqueness triples and amalgamation

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 [6]: (studying non-splitting) assuming the amalgamation property and tameness.In [7] we derive a good non-forking λ⁺-frame from a semi-good non-forking λ-frame. But the classes K_λ++ and {precedes above singleline equals sign}{up harpoon right} K_λ++ are replaced: K_λ++ is restricted to the saturated models and the partial order {precedes above singleline equals sign}{up harpoon right}K_λ++ is restricted to the partial order {precedes above singleline equals sign}λ+NF.Here, we avoid the restriction of the partial order {precedes above singleline equals sign}{up harpoon right}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 [7]): Theorem 7.15 states that M{precedes above singleline equals sign}M⁺ if and only if M{precedes above singleline equals sign}λ+NFM+, 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 [14], Vasey applies Theorem 7.8, proving the categoricity conjecture under the above 'unproven claim' of Shelah.In [10], we apply Theorem 7.15, proving the existence of primeness triples.