Graphs represented by Ext

Abstract

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 ℵ 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 ℵ 1 -free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.