Graphs represented by Ext

This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph $(μ,R)$ we can find a family $\{\mathbb G_α: α< μ\}$ of abelian groups such that for each $α,β\inμ$: $Ext(\mathbb G_α, \mathbb G_β) = 0 \Longleftrightarrow(α,β) \in R.$ 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 $λ= λ^{\aleph_0}$ and $μ= 2^λ$. We apply Jensen's diamond principle along with the criteria to present $λ$-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.

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