$\aleph_n$-free abelain group with no non-zero homomorphism to $\Bbb Z$

For any natural n, we construct an aleph_n-free abelian
groups which have few homomorphisms to Z . For this we use
``aleph_n-free (n+1)-dimensional black boxes''. The method is
relevant to e.g. construction of aleph_n-free abelian groups with
a prescribed endomorphism ring.

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