Extensions in the class of countable torsion-free Abelian groups

It is a classical result that for a torsion-free Abelian group A the group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname {Ext}_{\mathbb {Z}}(A,B)$\end{document} is divisible for any Abelian group B. Hence it is of the form [inline-graphic not available: see fulltext] for some uniquely determined cardinals r0 and rp. In this paper we clarify when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname {Ext}_{\mathbb {Z}}(A,B)=0$\end{document} and examine the possible values for r0 and rp in case the groups A and B are countable (torsion-free). We also give some methods for constructing torsion-free groups A and B with prescribed cardinals r0 and rp. This is to say that for suitable sequences (r0,rp∣p∈ℙ) of cardinals we construct torsion-free countable Abelian groups A and B realizing r0 and rp as their invariants of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname {Ext}_{\mathbb {Z}}(A,B)$\end{document}.