Knaster and friends I: Closed colorings and precalibers

TL;DR

This paper investigates the productivity of strong chain conditions like the Knaster property, providing ZFC examples where certain posets with precaliber properties fail to maintain these properties under infinite products, and studies related colorings.

Contribution

It constructs ZFC examples of posets with precaliber $\kappa$ whose powers are not $\kappa$-cc, and systematically studies colorings with strong unboundedness and closure properties.

Findings

Existence of ZFC examples of posets with precaliber $\kappa$ whose $\omega$-th power is not $\kappa$-cc.

Conditions under which strong unbounded colorings exist, especially when they are closed.

Analysis of the relationship between colorings and the productivity of chain conditions.

Abstract

The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares are not $\kappa$-cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, $\mathsf{ZFC}$ examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph_2$, 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 $\kappa$, we produce a $\mathsf{ZFC}$ example of a poset with precaliber $\kappa$ whose $\omega^{\mathrm{th}}$ power is not $\kappa$-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.