Graphs represented by Ext

Abstract This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\mu,R)} we can find a family <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>𝔾</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>&lt;</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{\mathbb{G}_{\alpha}:\alpha&lt;\mu\}} of abelian groups such that for each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>μ</m:mi> </m:mrow> </m:math> {\alpha,\beta\in\mu} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Ext</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>𝔾</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>𝔾</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {{\rm Ext}(\mathbb{G}_{\alpha},\mathbb{G}_{\beta})=0} iff <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mi>R</m:mi> </m:mrow> </m:math> {(\alpha,\beta)\in R} . In this regard, we present four results. First, we give a connection to Quillen’s small object argument which helps <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Ext</m:mi> </m:math> {{\rm Ext}} vanishes and use it to present a useful criteria to the question. Suppose <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>λ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>λ</m:mi> <m:msub> <m:mi mathvariant="normal">ℵ</m:mi> <m:mn>0</m:mn> </m:msub> </m:msup> </m:mrow> </m:math> {\lambda=\lambda^{\aleph_{0}}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>μ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mn>2</m:mn> <m:mi>λ</m:mi> </m:msup> </m:mrow> </m:math> {\mu=2^{\lambda}} . 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℵ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {\aleph_{1}} -free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.