Quite free complicated abelian groups, pcf and black boxes

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵ_ω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ₁-free, recently for ℵ_n-free). We succeed to prove the existence of even ℵω1⋅n-free ones. This requires building n-dimensional black boxes, which are quite free. This 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 ℵω1⋅ω-free Abelian group has non-trivial homomorphisms to ℤ.