Partition properties on Pκλ

Abstract

Menas [13] showed there exist normal ultrafilters on ${\mathcal{P}}_{\kappa }\lambda$ with the partition property if $\kappa$ is -supercompact. We first show that $\lambda$-supercompactness of $\kappa$ implies the existence of a normal ultrafilter on ${\mathcal{P}}_{\kappa }\lambda$ with the partition property. We also prove by a similar technic that part*$(\kappa ,$$\lambda )$ holds if $\kappa$ is $\lambda$-ineffable with $cf\left(\lambda \right)\ge \kappa$. Note that Magidor [11] showed $\kappa$ is $\lambda$-ineffable if part*$(\kappa ,$$\lambda )$ holds, and we proved the converse under some additional assumption in [7].