Partition properties on \mathscr{P}<SUB>κ</SUB>λ

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  • Partition properties on Pκλ
  • Partition properties on P kl

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Abstract

Menas [{13}] showed there exist 2^{2^{λ<κ}} normal ultrafilters on \mathscr{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 \mathscr{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)≥qκ. 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].

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