Uniqueness triples from the diamond axiom
We work with a $\lambda$-frame, which is an abstract elementary class endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality $\lambda$.
We will show that assuming the diamond axiom, any basic type admits a non-forking extension that has a uniqueness triple.
Prior results of Shelah in this direction required either some form of diamond at two consecutive cardinals, or a constraint on the number of models of size $\lambda$.
This is joint work of with Adi Jarden.