Some stationary subsets of $\mathcal {P}(\lambda )$

Abstract:

Let $\kappa$ and $\lambda$ be uncountable cardinals such that $\kappa \leq \lambda$, and set $S(\kappa ,\lambda ) = \left \{ {X \in {\mathcal {P}_\kappa }(\lambda )|\;|X \cap \kappa | < |X|} \right \}$. We determine the consistency strength of the statement "$\left ( {\exists \lambda \geq \kappa } \right )$($(S(\kappa ,\lambda )$ is stationary in ${\mathcal {P}_\kappa }(\lambda )$)" using a new type of partition cardinals. In addition, we show that the property "$S(\kappa ,{\kappa ^ + })$ is stationary in ${\mathcal {P}_\kappa }({\kappa ^ + })$" is much stronger.