Graphs represented by Ext

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 $\alpha ,\beta \in \mu$: $$Ext({\mathbb{G}}_{\alpha },{\mathbb{G}}_{\beta })=0⟺(\alpha ,\beta )\in R.$$ 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.