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Abstract

We investigate the problem of when ${\leq}\lambda$-support iterations of ${<}\lambda$-complete notions of forcing preserve $\lambda^+$. We isolate a property— properness over diamonds—that implies $\lambda^+$ is preserved and show that this property is preserved by $\lambda$-support iterations. Our condition is a relative of that presented by Rosłanowski and Shelah in \cite{RoSh:655}; 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 $\{\delta<\lambda^+:\mathop{\rm cf}(\delta)=\lambda\}$ that complements a theorem of Shelah \cite{Sh:f}.