I would like to know if a strengthening of the delta system lemma is true.
Suppose $\kappa$ is an infinite cardinal with $\kappa^{< \kappa} = \kappa$ (so $\kappa$ is regular). Suppose $S \subseteq \kappa^{+}$ is stationary. Let $\langle A_{\alpha} : \alpha \in S \rangle$ be such that each $A_{\alpha}$ is a subset of $\kappa^{+}$ of cardinality less than $\kappa$. Then I know (from Kunen's book) that there is some $W \subseteq S$ of cardinality $\kappa^{+}$ such that $\langle A_{\alpha} : \alpha \in W \rangle$ forms a delta system.
My question is: Does there exist some $W \subseteq S$, $\textbf{stationary}$ in $\kappa^{+}$ such that $\langle A_{\alpha} : \alpha \in W \rangle$ forms a delta system?
Supposing the above if false, I would also like to know if we can do this in the case if every member of $S$ has cofinality $\lambda$ for some fixed $\lambda < \kappa$.
Thanks!
Let $S = S^{\kappa^{+}}_{< \kappa} = \{\delta < \kappa^{+} : cf(\delta) < \kappa\}$. For $\delta \in S$, let $A_{\delta}$ be a club in $\delta$ of order type $cf(\delta)$ (so $|A_{\delta}| < \kappa$). Then by Fodor's lemma, for every stationary $W \subseteq S$, $\langle A_{\delta} : \delta \in W \rangle$ is not a delta system.
However, if $S \subseteq S^{\kappa^{+}}_{\kappa} = \{\delta < \kappa^{+} : cf(\delta) = \kappa\}$ is stationary then this is true by Fodor's lemma.