Forcing many positive polarized partition relations between a cardinal and its powerset

A fairly quotable special, but still representative, case of our main result is that for 2 ≤ n < ω, there is a natural number m(n) such that, the following holds. Assume GCH: If λ < μ are regular, there is a cofinality preserving forcing extension in which 2^λ = μ and, for all σ < λ ≤ κ < η such that η^(+m(n)-1) ≤ μ. ((η^(+m(n)-1))_σ) → ((κ)_σ)_η⁽¹⁾ⁿ. This generalizes results of [3], Section 1, and the forcing is a "many cardinals" version of the forcing there.