Density of uniqueness triples from the diamond axiom

TL;DR

This paper proves that in certain abstract elementary classes, the set of uniqueness triples is dense under weaker assumptions than previously required, advancing understanding of non-forking extensions.

Contribution

It significantly relaxes the hypotheses needed to establish the density of uniqueness triples in pre-$oldsymbol{ extlambda}$-frames, requiring only amalgamation, stability, and a diamond principle.

Findings

Density of uniqueness triples holds under weaker conditions.

Application to trivial $oldsymbol{ extlambda}$-frames shows density in any AEC with modest hypotheses.

Density of uniqueness triples in non-splitting relations with minimal assumptions.

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 $\diamondsuit(\lambda^+)$. 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 $*$-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.