More on the revised GCH and the black box

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