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Partitions and filters

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.