For an infinite cardinal $\lambda$ and stationary subset $S\subseteq\lambda^+$, why does $\Diamond^*_S\Rightarrow \Diamond_S$?
We use the notation from page $127$ of Assaf Rinot, Jensen’s diamond principle and its relatives, in Contemporary Mathematics $533$, Set Theory and Its Applications, Annual Boise Extravaganza in Set Theory, Boise, Idaho, $1995$-$2010$, L. Babinkostova, A.E. Caicedo, S. Geschke, M. Scheepers, eds., AMS, $2011$, which can be found here [PDF]:
$\lozenge_S$ asserts that there exists a sequence $\langle A_\alpha\mid\alpha\in S\rangle$ such that:
- for all $\alpha\in S$, $A_\alpha\subseteq\alpha$;
- if $Z$ is a subset of $\lambda^+$, then the following set is stationary: $$\{\alpha\in S\mid Z\cap\alpha=A_\alpha\}\;.$$
$\lozenge_S^*$ asserts that there exists a sequence $\langle\mathcal{A}_\alpha\mid\alpha\in S\rangle$ such that:
- for all $\alpha\in S$, $\mathcal{A}_\alpha\subseteq\mathcal{P}(\alpha)$ and $|\mathcal{A}_\alpha|\le\lambda$;
- if $Z$ is a subset of $\lambda^+$, then there exists a club $C\subseteq\lambda^+$ such that: $$C\cap S\subseteq\{\alpha\in S\mid Z\cap\alpha\in\mathcal{A}_\alpha\}\;.$$
Assume $\diamondsuit^{\star}_S$.
Using a bijection between $ \lambda^{+}$ and $\lambda \times \lambda^{+}$, first show that there exists $\langle \mathcal{B}_i : i \in S\rangle$ such that $\mathcal{B}_i = \{ B_{i, j} : j < \lambda \}$ where each $ B_{i, j} \subseteq \lambda \times i$ and for every $B \subseteq \lambda \times \lambda^{+}$, there is a club $C \subseteq \lambda^{+}$ (or just stationary $C \subseteq S$) such that for all $i \in C \cap S$, $B \cap (\lambda \times i) \in \mathcal{B}_i$.
Now let $B_{i, j, k} \subseteq i$ be the $k$-th column of $B_{i, j}$. Show that for some $j < \lambda$, $\langle B_{i, j, j} : i \in S \rangle$ is a $\diamondsuit_S$ witnessing sequence.