# Sh:1217

• Asgharzadeh, M., Golshani, M., & Shelah, S. Graphs represented by Ext. Preprint. arXiv: 2110.11143
• Abstract:
This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph (\mu,R) we can find a family \{\mathbb G_\alpha: \alpha < \mu\} of abelian groups such that for each \alpha,\beta\in\mu

Ext(\mathbb G_\alpha, \mathbb G_\beta) = 0 iff (\alpha,\beta) \in R.

In this regard, we present four results. First, we give a connection to Quillen’s small object argument which helps Ext vanishes and uses to present useful criteria to the question. Suppose \lambda = \lambda^{\aleph_0} and \mu = 2^\lambda. We apply Jensen’s diamond principle along with the criteria to present \lambda-free abelian groups representing bipartite graphs. Third, we use a version of the black box to construct in ZFC, a family of \aleph_1-free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.

• Version 2022-06-06 (48p)
Bib entry
@article{Sh:1217,
}