by Shelah. [Sh:486]

For a given stationary set S of countable ordinals we prove (in ZFC) the equivalence of a set theoretic statement and an abelian group theoretic statement. The first is ``every S-ladder system has aleph_0-uniformization''. The second says that every aleph_1-free (abelian) group of cardinality aleph_1 with non freeness invariant subseteq S is a Whitehead group and is even co-separable (i.e. Ext (G, Z)^omega =0); if S is not co-stationary the family of groups is somewhat more restricted. But only very elementary knowledge of abelian group theory and set theory is required. This solves problems from Eklof and Mekler's book.

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