## 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.

Original language | English |
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Article number | 90 |

Journal | Algebra Universalis |

Volume | 79 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2018 |

### Bibliographical note

Funding Information:Presented by A. Dow. This research was partially supported by the Israel Science Foundation (grant #1630/14).

Publisher Copyright:

© 2018, Springer Nature Switzerland AG.

## Keywords

- Closed coloring
- Knaster
- Precaliber
- Square
- Stationary reflection
- Unbounded function