Splitting stationary sets in $\mathscr{P}(\lambda)$

Abstract

Let $A$ be a non-empty set. $A$ set $S \subseteq \mathscr{P}(A)$ is said to be stationary in $\mathscr{P}(A)$ if for every $f: [A]^{< \omega} \to A$ there exists $ x \in S$ such that $x \neq A$ and $f"[x]^{<\omega}$. In this paper we prove the following: For an uncountable cardinal ? and a stationary set S in $\mathscr{P}(\lambda)$, if there is a regular uncountable cardinal $k \leq \lambda$ such that $\{x \in S: x \cap k \in k\}$ is stationary, then S can be split into $k$ disjoint stationary subsets.