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 {G_α : α < μ} of abelian groups such that for each α, β ∈ μ, Ext(G_α, G_β) = 0 iff (α, β) ∈ R. In this regard, we present four results. First, we give a connection to Quillen’s small object argument which helps Ext vanishes and use it to present a useful criteria to the question. Suppose λ = λ^ℵ0 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 ℵ₁-free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.