Some filters of partitions

Abstract

§0. Introduction. We started our study of filters of partitions in [15]. We shall here restrict ourselves to the consideration of filters on (ω)ω, the set of all infinite partitions of ω. §1 is an attempt to elucidate the connection between filters on (ω)ω and filters over ω. Given a filter H over ω, we define two filters FH and GH on (ω)ω, and we characterize p-points, rare ultrafilters and Ramsey ultrafilters in terms of properties of the associated filters of partitions.The remainder of the paper is devoted to the study of those filters that can be associated with Hindman's theorem and its extensions. Let us introduce some notation. Suppose * is an associative operation on ω, and let a subset A of ω and an ordinal α with 0 < α ≤ ω be given. We define a collection of subsets of ω by letting iff , where , is an increasing sequence of elements of A for each i < α, and whenever 0 < i < α. Then the Milliken-Taylor theorem (see [17] and [21]) asserts that for every F: [ω/n → m, where n and m are positive integers, there exists A Є [ω]ω such that F is constant on . Hindman's theorem [8] is the special case of this result when n = 1. (We shall conform to usage and simply write FS(A) instead of Glazer (see [6]) has given a proof of Hindman's theorem that uses idempotent ultrafilters. We shall see in §5 that the Milliken-Taylor theorem too can be derived in this fashion.