A fairly quotable special, but still representative, case of our main result is that for 2 $\leq$ n $\leq \omega$, there is a natural number m (n) such that, the following holds. Assume GCH: If $\lambda < \mu$ are regular, there is a cofinality preserving forcing extension in which 2$^\lambda = \mu$ and, for all $\sigma < \lambda \leq \kappa < \eta$ such that $\eta^{+m(n)-1)} \leq \mu$, $((\eta^{+m(n)-1)})_\sigma) \rightarrow ((\kappa)_\sigma)_\eta^{(1)n}.$ This generalizes results of [3], Section 1, and the forcing is a "many cardinals" version of the forcing there.