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The linear refinement number and selection theory

Volume 234 / 2016

Michał Machura, Saharon Shelah, Boaz Tsaban Fundamenta Mathematicae 234 (2016), 15-40 MSC: Primary 03E17; Secondary 03E75. DOI: 10.4064/fm124-8-2015 Published online: 17 February 2016

Abstract

The linear refinement number $\mathfrak {lr}$ is the minimal cardinality of a centered family in ${[\omega ]^{\omega }}$ such that no linearly ordered set in $({[\omega ]^{\omega }},\subseteq ^*)$ refines this family. The linear excluded middle number $\mathfrak {lx}$ is a variation of $\mathfrak {lr}$. 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 $\mathfrak {lr}=\mathfrak {lx}=\mathfrak {d}$ in all models where the continuum is at most $\aleph _2$, and that the cofinality of $\mathfrak {lr}$ is uncountable. Using the method of forcing, we show that $\mathfrak {lr}$ and $\mathfrak {lx}$ are not provably equal to $\mathfrak {d}$, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

Authors

  • Michał MachuraInstitute of Mathematics
    University of Silesia
    Bankowa 14
    40-007 Katowice, Poland
    and
    Department of Mathematics
    Bar-Ilan University
    Ramat Gan 5290002, Israel
    e-mail
  • Saharon ShelahEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Givat Ram, 9190401 Jerusalem, Israel
    and
    Mathematics Department
    Rutgers University
    New Brunswick, NJ, U.S.A.
    e-mail
  • Boaz TsabanDepartment of Mathematics
    Bar-Ilan University
    Ramat Gan 5290002, Israel
    and
    Department of Mathematics
    Weizmann Institute of Science
    Rehovot 7610001, Israel
    e-mail

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