More on the revised GCH and the black box

We strengthen the revised GCH theorem by showing, e.g., that for λ=cf(λ) >]_ω for all but finitely many regular k<]_ω it holds that " λ is accessible on cofinality k" in some weak sense (see below). As a corollary, λ=2^μ =μ⁺>]_ω implies that the diamond holds on λ when restricted to cofinality k for all but finitely many k ∈ Reg ∩ ]_ω We strengthen previous results on the black box and the middle diamond: previously it was established that these principles hold on { δ:δ<λ, cf (δ)=(] _n )⁺ } for sufficiently large n; here we succeed in replacing a sufficiently large ] _n with a sufficiently large א _n. The main theorem, concerning the accessibility of λ on cofinality K , Theorem 3.1, implies as a special case that for every regular λ > ] _ω , for some K <] _ω , we can find a sequence (P _δ : δ < λ) such that u ∈ P _δ ⇒ u = δ &Engl | u | < ] _ω , | P_δ | < λ, and we can fix a finite set d of "exceptional" regular cardinals θ < ] _ω so that if A ⊂λ satisfies | A | < ]_ω , there is a pair-coloring c : [ A ]² → k so that for every c-monochromatic B ∈ A with no last element, letting λ : = sup B it holds that B ∈ P_δ - provided that θ : = cf ( λ ) is not one of the finitely many "exceptional" members of d.