Tameness, Uniqueness 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: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking $\lambda^+$-frame from a semi-good non-forking $\lambda$-frame. But the classes $K_{\lambda^+}$ and $\preceq \restriction K_{\lambda^+}$ are replaced: $K_{\lambda^+}$ is restricted to the saturated models and the partial order $\preceq \restriction K_{\lambda^+}$ is restricted to the partial order $\preceq^{NF}_{\lambda^+}$. Here, we avoid the restriction of the partial order $\preceq \restriction K_{\lambda^+}$, assuming that every saturated model (in $\lambda^+$ over $\lambda$) is an amalgamation base and $(\lambda,\lambda^+)$-tameness for non-forking types over saturated models, (in addition to the hypotheses of [JrSh875]): We prove that $M \preceq M^+$ if and only if $M \preceq^{NF}_{\lambda^+}M^+$, provided that $M$ and $M^+$ are saturated models. We present sufficient conditions for three good non-forking $\lambda^+$-frames: one relates to all the models of cardinality $\lambda^+$ and the two others relate to the saturated models only. By an `unproven claim' of Shelah, if we can repeat this procedure $\omega$ times, namely, `derive' good non-forking $\lambda^{+n}$ frame for each $n<\omega$ then the categoricity conjecture holds. Vasey applies one of our main theorems in a proof of the categoricity conjecture under the above `unproven claim' of Shelah and more assumptions. In [Jrprime], we apply the main theorem in a proof of the existence of primeness triples.