Non-forking w-good frames

Abstract

We introduce the notion of a w-good $\lambda$-frame which is a weakening of Shelah's notion of a good $\lambda$-frame. Existence of a w-good $\lambda$-frame implies existence of a model of size ${\lambda }^{++}$. Tameness and amalgamation imply extension of a w-good $\lambda$-frame to larger models. As an application we show: $\text{Theorem}$ Suppose $2^{\lambda }<2^{{\lambda }^{+}}<2^{{\lambda }^{++}}$ and $2^{{\lambda }^{+}}>{\lambda }^{++}$. If $I(K,\lambda )=I(K,{\lambda }^{+})=1\le I(K,{\lambda }^{++})<2^{{\lambda }^{++}}$ and $K$ is $(\lambda ,{\lambda }^{+})$-tame, then $K_{{\lambda }^{+++}}\ne \varnothing$. The proof presented clarifies some of the details of the main theorem of [Sh576] and avoids using the heavy set-theoretic machinery of [Sh: h §VII] by replacing it with tameness.

22 pages; 2 figures; fixed a few typos

References in corpus (2)

• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I

Cited by in corpus (1)

• Density of uniqueness triples from the diamond axiom