Abstract Let $\kappa $ be a regular uncountable cardinal, and a cardinal greater than or equal to $\kappa $ . Revisiting a celebrated result of Shelah, we show that if is close to $\kappa $ and (= the least size of a cofinal subset of ) is greater than , then can be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that if and , then no $\kappa $ -complete ideal on is weakly -saturated.

Abstract Let κ be a regular uncountable cardinal and λ be a cardinal greater than κ. We show that if 2<κ ≤ M(κ, λ), then ◇κ,λ holds, where M(κ, λ) equals λℵ0 if cf(λ) ≥ κ, and (λ+)ℵ0 otherwise.

Abstract We study the partition relation that is a weakening of the usual partition relation . Our main result asserts that if κ is an uncountable strongly compact cardinal and , then does not hold.

Abstract We study the degree of (weak) (Q-pointness, and that of (weak) P-pointness, of ideals on a regular infinite cardinal.

Abstract Let P be any one of the following combinatorial properties: weak P-pointness, weak (semi-) Q-pointness, weak (semi-)selectivity, ω-closedness. We deal with the following two questions: (1) What is the least cardinal k such that there exists an ideal with k many generators that does not have the property P? (2) Can one extend every ideal with the property P to a prime ideal with the property P?

§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.

In [2], Carlson and Simpson proved a dualized version of Ramsey's theorem obtained by coloring partitions of ω instead of subsets of ω. It was at the suggestion of Simpson that the author undertook to study the notion dual to that of a Ramsey ultrafilter. After stating the basic terminology and notation used in the paper in §1, in §2 we establish some basic properties of the lattice of all partitions of a cardinal κ. §3 is devoted to the study of families of pairwise disjoint partitions of ω. §4 is concerned with descending sequences of partitions. In §5, we give some examples of filters of partitions. Properties of such filters are discussed in §6. Co-Ramsey filters are introduced in §7, and it is shown how they can be associated with Ramsey ultrafilters. The main result of §8 is Proposition 8.1, which asserts the existence of a co-Ramsey filter under the continuum hypothesis. We use standard set theoretic conventions and notation. Let κ be a cardinal. We set κ* = κ − {0}. For every ordinal α ≤ κ, (κ)α denotes the set of those sequences X(ν), ν < α, of pairwise disjoint nonempty subsets of κ such that ⋃ν<αX(ν) = κ, and ⋂X(ν) < ⋂X(ν′) whenever ν < ν′. We also let (κ)≤α = ⋃β≤α(κ)β and (κ)<α = ⋃β<α(κ)β. Given X ∈ (κ)α, we put xν = ⋂X(ν) for every ν < α, and we denote by Ax the set of all xν, 0 < ν < α.