A Note on a Result of Kunen and Pelletier

Suppose that U and U' are normal ultrafilters associated with some supercompact cardinal. How may we compare U and U'? In what ways are they similar, and in what ways are they different? Partial answers are given in [1], [2], [3], [5], [6], and [7]. In this paper, we continue this study. In [6], Menas introduced a combinatorial principle X(U) of normal ultrafilters U associated with supercompact cardinals, and showed that normal ultrafilters satisfying this property also satisfy a partition property. In [5], Kunen and Pelletier showed that this partition property for U does not imply X(U). Using results from [3], we present a different method of finding such normal ultrafilters which satisfy the partition property but do not satisfy x(U). Our method yields a large collection of such normal ultrafilters. ?1. Preliminaries. We assume the reader is familiar with the basic notation, definitions, and techniques involving supercompact cardinals (see [6]). For K A. K is supercompact iff K is A-supercompact for all A ? K. For U a normal ultrafilter on PK({), let iu: V -+Mu denote the canonical corresponding embedding and inner model. Suppose K : X eA, Ye A, and X i Y}] = {i}. A is said to be homogeneous for the partition F. Received January 29, 1991. ? 1992, Association for Symbolic Logic 0022-4812/92/5702-0006/$01 .50