The linear refinement number and selection theory

Michał Machura, Saharon Shelah, Boaz Tsaban

Research output: Contribution to journalArticlepeer-review


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
Issue number1
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2016.


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


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