Density of uniqueness triples from the diamond axiom

Abstract

We work with a pre-$\lambda$-frame, which is an abstract elementary class (AEC) endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality $\lambda$. We investigate the density of uniqueness triples in a given pre-$\lambda$-frame $\mathfrak{s}$, that is, under what circumstances every basic triple admits a non-forking extension that is a uniqueness triple. Prior results in this direction required strong hypotheses on $\mathfrak{s}$. Our main result is an improvement, in that we assume far fewer hypotheses on $\mathfrak{s}$. In particular, we do not require $\mathfrak{s}$ to satisfy the extension, uniqueness, stability, or symmetry properties, or any form of local character, though we do impose the amalgamation and stability properties in ${\lambda }^{+}$, and we do assume $♢({\lambda }^{+})$. As a corollary, by applying our main result to the trivial $\lambda$-frame, it follows that in any AEC $\mathbf{K}$ satisfying modest hypotheses on ${\mathbf{K}}_{\lambda }$ and ${\mathbf{K}}_{{\lambda }^{+}}$, the set of $\ast$-domination triples in ${\mathbf{K}}_{\lambda }$ is dense among the non-algebraic triples. We also apply our main result to the non-splitting relation, obtaining the density of uniqueness triples from very few hypotheses.

Expanded with more corollaries since v2; now 37 pages