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.

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相当自由的复杂阿贝尔群、pcf 和黑盒

我们喜欢构建阿贝尔群（或 R 模），一方面是非常自由的，比如 $\aleph_{\omega + 1}$-free，另一方面，在适当的意义上是复杂的。我们选择对 $Z$ 没有非平凡同态作为我们的测试问题（经典地以 $\aleph_1$-free 着称，最近以 $\aleph_n$-free 着称）。我们成功地证明了甚至 $\aleph_{\omega_1 \cdot n}-$free 的存在。这需要构建非常免费的 n 维黑匣子。因此，组合学是出于自身利益，我们相信也可用于其他目的。另一方面，取模合适的大基数，我们证明每个 $\aleph_{\omega_1 \cdot \omega}$-free 阿贝尔群对 Z 具有非平凡同态是一致的。