Knaster and friends I: closed colorings and precalibers

Abstract

The productivity of the κ-chain condition, where κ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of κ-cc posets whose squares are not κ-cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which κ = ℵ 2 , was resolved by Shelah in 1997.In this work, we obtain analogous results regarding the infinite productivity of strong chain conditions, such as the Knaster property. Among other results, for any successor cardinal κ, we produce a ZFC example of a poset with precaliber κ whose ω th power is not κ-cc. To do so, we carry out a systematic study of colorings satisfying a strong unboundedness condition. We prove a number of results indicating circumstances under which such colorings exist, in particular focusing on cases in which these colorings are moreover closed.

Knaster and friends II: The C-sequence number

“…), starting from a large cardinal notion weaker than strong compactness. It is analogous to Theorem 2.14 of [15]. Theorem 3.10.…”

“…Working in V [G], fix θ ∈ Reg(λ + )\{cf(λ), θ} and a coloring c : [λ + ] 2 → θ . We will find a family A ⊆ [λ + ] ≤θ consisting of λ + -many pairwise disjoint sets and an ordinal k < θ such that, for all (a, b) ∈ [A] 2 , we have min(c[a × b]) ≤ k. For all i < θ, forcing with T i resurrects the fact that λ is a singular limit of strongly compact cardinals, and so by [15,Theorem 2.14…”

“….). Before getting to those connections, let us recall some relevant definitions and results from [15]. 1 ([15]).…”

“…By an analysis from the forthcoming Part III of this series, in the model of Theorem 3.4, in which χ(κ) = ω, there is a closed witness to U(κ, κ, ω, κ). Also, by Theorem 2.14, if λ is a limit of an ω-sequence of strongly compact cardinals, then χ(λ + ) = ω, but, by [15,Corollary 4.13], there is a closed witness to U(λ + , λ + , ω, λ).…”

“…Proof. By [15,Corollary 4.10], the hypothesis implies that, for every θ ∈ Reg(κ), there exists a closed witness to U(κ, κ, θ, χ). Now, appeal to (2) ⇒ (1) of Corollary 5.21.…”