Hierarchies of ineffabilities

We study combinatorial large cardinal properties on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {P}}_{\kappa } \lambda$\end{document} , such as ineffability, almost ineffability, subtlety, and the Shelah property. We show that, even when λ > κ, the almost ineffability of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {P}}_{\kappa } \lambda$\end{document} does not yield the ineffability of κ. We also show that the Shelah property and the partition property of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {P}}_{\kappa } \lambda$\end{document} do not yield the subtlety of κ.