Algebra universalis

, 79:90 | Cite as

Knaster and friends I: closed colorings and precalibers

  • Chris Lambie-HansonEmail author
  • Assaf Rinot


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.


Knaster Precaliber Closed coloring Unbounded function Stationary reflection Square 

Mathematics Subject Classification

03E35 03E05 03E75 06E10 



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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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael
  2. 2.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA

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