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We like to build Abelian groups (or R-modules) which on the one hand arequite free, say $\aleph_{\omega + 1}$-free, and on the other hand, arecomplicated in suitable sense. We choose as our test problem having nonon-trivial homomorphism to $Z$ (known classically for $\aleph_1$-free,recently for $\aleph_n$-free). We succeed to prove the existence of even$\aleph_{\omega_1 \cdot n}-$free ones. This requires building n-dimensionalblack boxes, which are quite free. Thus combinatorics is of self interest andwe believe will be useful also for other purposes. On the other hand, modulosuitable large cardinals, we prove that it is consistent that every$\aleph_{\omega_1 \cdot \omega}$-free Abelian group has non-trivialhomomorphisms to Z.

Quite free complicated abelian group, PCF and Black Boxes