Embeddings of Cohen algebras (Q1355471)

Terminology: A poset \(P\) has \textit{uniform density} \(\kappa\) if \(|P|=\kappa\) and there do not exist \(p\in P\) and \(R\in[P]^{<\kappa}\) such that \(R\) is dense below \(p\). A Boolean algebra has \textit{uniform density} \(\kappa\) if it has a dense subset of uniform density \(\kappa\). For any infinite cardinal \(\kappa\), denote by \({\mathbb C}(\kappa)\) the algebra of regular open sets in \(\{0,1\}^\kappa\). Main theorem: If ZFC is consistent then so is ZFC \(+\) \(2^{\aleph_0}=\aleph_2\) \(+\) ``every complete Boolean algebra \({\mathfrak B}\) of uniform density \(\aleph_1\) has a complete subalgebra isomorphic to \({\mathbb C}(\aleph_1)\)''.