Every (λ, Κ)-regular Ultrafilter Is (λ, Κ)-regular

We prove the following: Theorem A. If D is a (λ+, κ)-regular ultrafilter, then either (a) D is (λ, κ)-regular, or (b) the cofinality of the linear order ∏ D〈λ, <〉 is cf κ, and D is (λ, κ′)-regular for all κ′ < κ. Corollary B. Suppose that κ is singular, κ > λ and either λ is regular, or cf κ < cf λ. Then every (λ+n, κ)-regular ultrafilter is (λ, κ)-regular. We also discuss some consequences and variations. The notion of a (λ, κ)-regular ultrafilter has been introduced by J. Keisler in [Kei]. An ultrafilter D is (λ, κ)-regular iff there is a family of κ members of D such that the intersection of any λ members of the family is empty. In [Kei] Keisler proved some cardinality results about ultraproducts taken modulo such ultrafilters. Further results were proved in the 70’s: for example, the following are theorems of ZFC: (a) Every (λ, λ)-regular ultrafilter is (λ, λ)-regular ([CC], [KP]). (b) If λ is singular, then every (λ, λ)-regular ultrafilter is (λ, λ)-regular [Ka]; moreover, it is either (cf λ, cf λ)-regular or (λ′, λ)-regular for some λ′ < λ ([CC], [KP]). (c) If 2κ = κ and 2κ+ > κ, then every (κ, κ)-regular ultrafilter is (κ, κ)-regular ([BK], [Ket]). It was soon realized, however, that (ir)regular ultrafilters are connected with large cardinals, inner models, and combinatorial or reflection principles; this paved the way for significant applications to set theory, but seemed to dash any hope that other results besides (a)–(c) above can be proved in ZFC alone (see e.g. [KM] or [Lp1] for further references). However, in [Lp1] (more than twenty years later) we proved some slight improvements of (b), as well as a “down from exponents” transfer result for (λ, λ)-regularity; we also suggested the possibility that further results are theorems of ZFC. It is actually so; in this paper we prove the following generalization of (a): Theorem 1. If n < ω, then every (λ, κ)-regular ultrafilter is (λ, κ)-regular. Received by the editors November 20, 1997 and, in revised form, April 8, 1998. 1991 Mathematics Subject Classification. Primary 03C20, 04A20.