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, \(\textsf {ZFC}\) 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 \(\textsf {ZFC}\) 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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Acknowledgements
The results of this paper were presented by the first author at the Set Theory, Model Theory and Applications conference in Eilat, April 2018, and at the SETTOP 2018 conference in Novi Sad, July 2018, and by the second author at the \(11^{\text {th}}\) Young Set Theory workshop in Lausanne, June 2018. We thank the organizers for the warm hospitality.
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Presented by A. Dow.
This research was partially supported by the Israel Science Foundation (grant #1630/14)
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Lambie-Hanson, C., Rinot, A. Knaster and friends I: closed colorings and precalibers. Algebra Univers. 79, 90 (2018). https://doi.org/10.1007/s00012-018-0565-1
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DOI: https://doi.org/10.1007/s00012-018-0565-1