Joint work with Chris Lambie-Hanson.
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, 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 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.
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Citation information:
C. Lambie-Hanson and A. Rinot, Knaster and friends I: closed colorings and precalibers, Algebra Universalis, 79(4): 90, 2018.
Submitted to Algebra Universalis, June 2018.
Accepted, September 2018.
Correction of two typos:
1. At the opening of the proof of Theorem 4.23, where it says “By Corollary 4.19”, it should have been “By Corollary 4.12”.
2. Later in the proof of Theorem 4.23, where defining the ordinal $\Lambda$, the function $\lambda_2^k$ should have been $\lambda_2$ (the extra superscript is only relevant to the proof of Case 2 of Theorem 4.21, and is redundant here).