  We prove in ZFC that an abelian group $C$ is cotorsion if and only if
$\operatorname{Ext}(F,C) = 0$ for every $\aleph_k$-free group $F$, and discuss
some consequences and related results. This short note includes a condensed
overview of the $\bar\lambda$-Black Box for $\aleph_k$-free constructions in
ZFC.

Group Theory;Logic