Menas  showed there exist normal ultrafilters on with the partition property if is -supercompact. We first show that -supercompactness of implies the existence of a normal ultrafilter on with the partition property. We also prove by a similar technic that part* holds if is -ineffable with . Note that Magidor  showed is -ineffable if part* holds, and we proved the converse under some additional assumption in .
"Partition properties on ." J. Math. Soc. Japan 54 (1) 123 - 133, January, 2002. https://doi.org/10.2969/jmsj/1191593957