Abstract: 

\def\cf{\mathop{\rm cf}}We investigate the problem of when ${\leq}\lambda$-support
iterations of ${<}\lambda$-comp\-le\-te notions of forcing preserve
$\lambda^+$. We isolate a property---{\em properness over
diamonds}\break---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
Ros³anowski 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
$\{\delta<\lambda^+:\cf(\delta)=\lambda\}$ that complements a
theorem of Shelah \cite{Sh:f}.

1. Department of Mathematics
   University of Northern Iowa
   Cedar Falls, IA 50614, U.S.A.