Graphs represented by Ext

Abstract This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph (μ, R) (, R) we can find a family 𝔾 α: α μ \{G_{: α, β ∈ μ, , Ext ⁢ (𝔾 α, 𝔾 β) = 0 { Ext (G_, G_) =0} iff (α, β) ∈ R (, ) R. In this regard, we present four results. First, we give a connection to Quillen’s small object argument which helps Ext { Ext} vanishes and use it to present a useful criteria to the question. Suppose λ = λ ℵ 0 =^{₀} and μ = 2 λ =2^{}. 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 ℵ 1 ₁ -free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.