More on the revised GCH and the black box

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