  We like to build Abelian groups (or R-modules) which on the one hand are
quite free, say $\aleph_{\omega + 1}$-free, and on the other hand, are
complicated in suitable sense. We choose as our test problem having no
non-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-dimensional
black boxes, which are quite free. Thus combinatorics is of self interest and
we believe will be useful also for other purposes. On the other hand, modulo
suitable large cardinals, we prove that it is consistent that every
$\aleph_{\omega_1 \cdot \omega}$-free Abelian group has non-trivial
homomorphisms to Z.
