On Iterated Forcing for Successors of Regular Cardinals

We investigate the problem of when ≤ λ–support iterations of < λ–complete notions of forcing preserve λ. We isolate a property — properness over diamonds — that implies λ is preserved and show that this property is preserved by λ–support iterations. Our condition is a relative of that presented in [1]; it is not clear if the two conditions are equivalent. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on {δ < λ : cf(δ) = λ} that complements a theorem of Shelah in [3].