The linear refinement number and selection theory

Michał Machura ¹   ; Saharon Shelah ²   ; Boaz Tsaban ³

¹ Institute of Mathematics University of Silesia Bankowa 14 40-007 Katowice, Poland and Department of Mathematics Bar-Ilan University Ramat Gan 5290002, Israel
² Einstein Institute of Mathematics The Hebrew University of Jerusalem Givat Ram, 9190401 Jerusalem, Israel and Mathematics Department Rutgers University New Brunswick, NJ, U.S.A.
³ Department of Mathematics Bar-Ilan University Ramat Gan 5290002, Israel and Department of Mathematics Weizmann Institute of Science Rehovot 7610001, Israel

Fundamenta Mathematicae, Tome 234 (2016) no. 1, pp. 15-40

Résumé

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.

Détail

DOI : 10.4064/fm124-8-2015

Keywords: linear refinement number mathfrak minimal cardinality centered family omega omega linearly ordered set omega omega subseteq * refines family linear excluded middle number mathfrak variation mathfrak these numbers estimate critical cardinalities number selective covering properties compare these numbers classical combinatorial cardinal characteristics continuum prove mathfrak mathfrak mathfrak models where continuum aleph cofinality mathfrak uncountable using method forcing mathfrak mathfrak provably equal mathfrak rule out several potential bounds these numbers results solve number problems