A note on a lemma of Shelah concerning stationary sets

Abstract:

Let $\kappa$ be an infinite cardinal, let $I$ be a nonprincipal ideal on $\kappa$ and let ${I^ + } = \{ X \subseteq \kappa :X \notin I\}$. $S(I)$ is the following property of ideals: for every $A \in {I^ + }$ and every pair of functions $f,g$ from $A$ into $\kappa$ such that, for every $\alpha \in A$, $f(a) \ne g(\alpha )$, there exists a set $B \subseteq A$ with $B \in {I^ + }$ such that $f''B \cap g''B = \emptyset$. We prove that $S(I)$ holds for every weakly selective ideal $I$ on any infinite cardinal $\kappa$ (including $\kappa = \omega$), and that $S(I)$ holds for every $\kappa$-complete ideal on $\kappa$ iff $\kappa$ is not strongly inaccessible.