### Uniformization and the diversity of Whitehead groups

by Eklof and Mekler and Shelah. [EMSh:441]

Israel J Math, 1992

The connections between Whitehead groups and uniformization
properties were investigated by the third author in [Sh:98]. In
particular it was essentially shown there that there is a non-free
Whitehead (respectively, aleph_1-coseparable) group of
cardinality aleph_1 if and only if there is a ladder system on a
stationary subset of omega_1 which satisfies 2-uniformization
(respectively, omega-uniformization). These techniques allowed
also the proof of various independence and consistency results about
Whitehead groups, for example that it is consistent that there is a
non-free Whitehead group of cardinality aleph_1 but no non-free
aleph_1-coseparable group. However, some natural questions
remained open, among them the following two: (i) Is it consistent
that the class of W-groups of cardinality aleph_1 is exactly the
class of strongly aleph_1-free groups of cardinality aleph_1 ?
(ii) If every strongly aleph_1-free group of cardinality
aleph_1 is a W-group, are they also all aleph_1-coseparable?
In this paper we use the techniques of uniformization to answer the
first question in the negative and give a partial affirmative answer
to the second question.

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