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
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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\muExt(\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,
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}
}