Between Whitehead groups and uniformization

Abstract

For a given stationary set $S$ of countable ordinals we prove (in $\mathbf{ZFC}$) that the assertion "every $S$-ladder system has ${\aleph }_0$-uniformization" is equivalent to "every strongly ${\aleph }_1$-free abelian group of cardinality ${\aleph }_1$ with non-freeness invariant $\subseteq S$ is ${\aleph }_1$-coseparable, i.e. Ext$(G,{\oplus }_{i=0}^{\infty }\mathbb{Z})=0$ (in particular Whitehead, i.e.\ Ext$(G,\mathbb{Z})=0$)". This solves problems B3 and B4 from Eklof and Mekler's monograph.