Quite free complicated abelian group, PCF and Black Boxes

Abstract

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.