The linear refinement number and selection theory

Michał Machura, Saharon Shelah, Boaz Tsaban

Research output: Contribution to journalArticlepeer-review

Abstract

The linear refinement number lr is the minimal cardinality of a centered family in [ω]ω such that no linearly ordered set in ([ω]ω,⊆) refines this family. The linear excluded middle number lr is a variation of lr. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that lr = lŗ = d in all models where the continuum is at most ℵ2, and that the cofinality of lr is uncountable. Using the method of forcing, we show that lr and lŗ are not provably equal to d, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

Original languageEnglish
Pages (from-to)15-40
Number of pages26
JournalFundamenta Mathematicae
Volume234
Issue number1
DOIs
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2016.

Funding

We thank Daniel Hathaway for his comments on the proof of Theorem 2.10, and the referee for useful suggestions. A part of this material was presented at the Tel Aviv University Set Theory Seminar. We thank Moti Gitik and Assaf Rinot for their useful feedback. We are indebted to Lyubomyr Zdomskyy for reading the first version of this paper, detecting a large number of errors, and making many useful suggestions. The research of the first named author was partially supported by the Science Absorption Center, Ministry of Immigrant Absorption, the State of Israel. The research of the second named author was partially supported by the Israel Science Foundation, grant number 1053/11. This is publication 1032 of the second named author.

FundersFunder number
Science Absorption Center
Ministry of Aliyah and Immigrant Absorption
Israel Science Foundation1053/11
Tel Aviv University

    Keywords

    • Forcing
    • Linear refinement number
    • Mathias forcing
    • Pseudointersection number
    • Selection principles
    • γ-Cover
    • τ-Cover
    • τ-cover
    • ω-Cover

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