Forcing many positive polarized partition relations between a cardinal andits powerset

A fairly quotable special, but still representative, case of our main resul t is that for 2 less than or equal to n < omega, there is a natural number in (n) such that, the following holds. Assume GCH: If lambda < mu are regul ar, there is a cofinality preserving forcing extension in which 2(lambda) = mu and, for all sigma < lambda less than or equal to kappa < eta such that eta ((+m(n)-1)) less than or equal to mu, ((eta ((+m(n)-1)))(sigma)) --> ((kappa)(sigma))(eta)((1)n). This generalizes results of [3], Section 1, and the forcing is a "many card inals" version of the forcing there.