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+λ+-frame from a semi-good non-forking λ  -frame. But the classes Kλ+Kλ+ and ⪯↾Kλ+⪯↾Kλ+ are replaced: Kλ+Kλ+ is restricted to the saturated models and the partial order ⪯↾Kλ+⪯↾Kλ+ is restricted to the partial order ⪯λ+NF.Here, we avoid the restriction of the partial order ⪯↾Kλ+⪯↾Kλ+, assuming that every saturated model (in λ+λ+ over λ  ) is an amalgamation base and (λ,λ+)(λ,λ+)-tameness for non-forking types over saturated models (in addition to the hypotheses of [7]): Theorem 7.15 states that M⪯M+M⪯M+ if and only if M⪯λ+NFM+, provided that M   and M+M+ are saturated models.We present sufficient conditions for three good non-forking λ+λ+-frames: one relates to all the models of cardinality λ+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+nλ+n frame for each n<ωn<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.