Partition properties on P_κλ

摘要

Menas [13]showed there exit 2~2λ~＜κ normal ultrafilters on P_κλ with the partition property if κ is 2~λ＜κ -supercompact. We first show that λ-supercompactness of κ implies the existence of a normal ultrafilter on P_κλ with the partition property. We also prove by a similar technic that part (κ,λ) holds if κ is λ-ineffable with cf(λ)≤κ. Note that Magidor [11] showed κ is λ-ineffable if part (κ,λ) holds, and we proved the Converse under some additional assumption in [7].