Sh:1217

• Asgharzadeh, M., Golshani, M., & Shelah, S. (2025). Graphs represented by Ext. Forum Math., 38(1), 215–241. arXiv: 2110.11143 DOI: 10.1515/forum-2022-0170
• 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)

@article{Sh:1217,
 author = {Asgharzadeh, Mohsen and Golshani, Mohammad and Shelah, Saharon},
 title = {{Graphs represented by Ext}},
 journal = {Forum Math.},
 fjournal = {Forum Mathematicum},
 volume = {38},
 number = {1},
 year = {2025},
 pages = {215-241},
 mrclass = {03C60; 20A15; 13L05},
 doi = {10.1515/forum-2022-0170},
 note = {\href{https://arxiv.org/abs/2110.11143}{arXiv: 2110.11143}},
 arxiv_number = {2110.11143}
}