My paper The Pseudopower Dichotomy is in the latest issue of the Journal of Symbolic Logic. I like to think of the main theorem as a ZFC result in classical cardinal arithmetic that only becomes apparent when you use the apparatus of pcf theory to transform questions about the cofinality of structures consisting of sets ordered by inclusion to questions about the cofinality of sets of functions ordered modulo certain types of ideals. I've written down particular instances of the main theorem where the choice of parameters is for amusement purposes only...
If \(\mu\) is a singular cardinal of cofinality \(\aleph_6\) then
\({\rm cov}(\mu,\mu,\aleph_9,\aleph_2)={\rm cov}(\mu,\mu,\aleph_7,\aleph_2)+{\rm cov}(\mu,\mu,\aleph_9,\aleph_6)\)
These covering numbers are finer versions of the cofinality of \(([\mu]^{<\mu},\subseteq)\). For example \({\rm cov}(\mu,\mu,\aleph_9,\aleph_2)\) is the minimum cardinality of \(\mathcal{P}\subseteq [\mu]^{<\mu}\) with the property that any set in \([\mu]^{<\aleph_9}\) can be covered by a union of fewer than \(\aleph_2\) sets from \(\mathcal{P}\). This cardinal can be manipulated by forcing, but the content of the theorems says that increasing it means that you must increase at least one of the two numbers on the right hand side. What I found most interesting is that work of Gitik on the Shelah Weak Hypothesis shows that both terms on the right side of the equation are necessary.