Diamond, GCH and weak square

Abstract:

Shelah proved recently that if $\kappa >\omega$ and $S\subseteq \kappa ^+$ is a stationary set of ordinals of cofinality different from $\mathrm {cf}(\kappa )$, then $2^\kappa =\kappa ^+$ implies $\Diamond _{\kappa ^+}(S)$. We show that for singular $\kappa$, an elaboration on his argument allows us to derive $\Diamond _{\kappa ^+}(T)$ from $2^\kappa =\kappa ^+ + \square ^*_\kappa$ where $T=\{\delta <\kappa ^+ {|} \mathrm {cf}(\delta )=\mathrm {cf}(\kappa )\}$. This gives a strong restriction on the existence of saturated ideals on $\kappa ^+$.