More on the revised GCH and the black box

We strengthen the revised GCH theorem by showing, e.g., that for , for all but finitely many regular κ<ℶω, it holds that “λ is accessible on cofinality κ” in some weak sense (see below).As a corollary, λ=2μ=μ+>ℶω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many .We strengthen previous results on the black box and the middle diamond: previously it was established that these principles hold on for sufficiently large n; here we succeed in replacing a sufficiently large ℶn with a sufficiently large ℵn.The main theorem, concerning the accessibility of λ on cofinality κ, Theorem 3.1, implies as a special case that for every regular λ>ℶω, for some κ<ℶω, we can find a sequence 〈Pδ:δ<λ〉 such that , |Pδ|<λ, and we can fix a finite set d of “exceptional” regular cardinals θ<ℶω so that if A⊆λ satisfies |A|<ℶω, there is a pair-coloring so that for every -monochromatic B⊆A with no last element, letting δ:=supB it holds that B∈Pδ—provided that is not one of the finitely many “exceptional” members of d.