The linear refinement number and selection theory
Volume 234 / 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.
