Tameness and joint embedding for the Paolini–Shelah torsion-free abelian-group AEC

Determine whether the abstract elementary class of torsion-free abelian groups constructed by Paolini and Shelah (arXiv:2512.02545v1, Section 4), equipped with the pure subgroup relation, is (<aleph_0)-tame and whether it satisfies the joint embedding property, in order to clarify the structural behavior of that counterexample to stability.

Background

Paolini and Shelah constructed an abstract elementary class of torsion-free abelian groups witnessing that not all AECs of modules with pure embeddings are stable. The present paper develops a variation of that construction which is shown to have the joint embedding property and to be (<aleph_0)-tame.

However, for the original Paolini–Shelah example, the authors explicitly note that it is unknown whether it enjoys these two properties. Establishing the status of (<aleph_0)-tameness and the joint embedding property for the Paolini–Shelah class would clarify how pervasive these good structural features are in such instability examples.