The Partition Property for Certain Extendible Measures on Supercompact Cardinals

Abstract

We give an alternate characterization of a combinatorial property of measures on ${p_\kappa }\lambda$ introduced by Menas. We use this characterization to prove that if $\kappa$ is supercompact, then all measures on ${p_\kappa }\lambda$ in a certain class have the partition property. This result is applied to obtain a self-contained proof that if $\kappa$ is supercompact and $\lambda$ is the least measurable cardinal greater than $\kappa$, then Solovay’s "glue-together" measures on ${p_\kappa }\lambda$ are not ${2^\lambda }$-extendible.