On a Combinatorial Property of Menas Related to the Partition Property for Measures on Supercompact Cardinals

Abstract

T. K. Menas [4, pp. 225-234] introduced a combinatorial property $\chi (\mu)$ of a measure $\mu$ on a supercompact cardinal $\kappa$ and proved that measures with this property also have the partition property. We prove here that Menas' property is not equivalent to the partition property. We also show that if $\alpha$ is the least cardinal greater than $\kappa$ such that $P_\kappa\alpha$ bears a measure without the partition property, then $\alpha$ is inaccessible and $\Pi^2_1$-indescribable.