Normal ultrafilters without the partition property
by
Shizuo Kamo
Osaka Prefecture University

Let \kappa be a measurable cardinal and \kappa <= \lambda. Concerning the partition property of a normal ultrafilter on P_\kappa\lambda, Solovay proved the existence of a normal ultrafilter without the partition property under the assumption of that the existence of a certain large cardinal greater than \kappa. After Solovay established this result, Kunen improved his results, and proved that the existence of a normal ultrafilter without the partition property implies the existence of a certain large cardinal above \kappa. On the other hand, Menas proved that there exist 2^{2\lambda < \kappa} normal ultrafilters with the partition property, if \kappa is 2^{\lambda < \kappa} supercompact. In the talk, we prove

Theorem Suppose that U is a normal ultrafilter on P_\kappa\lambda without the partition property.

Define \theta by

UltU(V) \models ''\theta is the first Mahlo cardinal greater than \lambda''.

Then, it holds that

UltU(V) \models''\kappa is \gamma-supercompact for all\gamma < \theta''.

As a corollary, we have the following which has been proved.

Corollary If \kappa is \lambda-supercompact, then there exists a normal ultarfilter on P_\kappa\lambda with the partition property.

Date received: February 9, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagb-05.