The linear refinement number and selection theory

Michał Machura, Saharon Shelah, and Boaz Tsaban. "The linear refinement number and selection theory." Fundamenta Mathematicae 234.1 (2016): 15-40. <http://eudml.org/doc/286432>.

@article{MichałMachura2016,
abstract = {The linear refinement number 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 is a variation of . 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 = = in all models where the continuum is at most ℵ₂, and that the cofinality of is uncountable. Using the method of forcing, we show that and are not provably equal to , and rule out several potential bounds on these numbers. Our results solve a number of open problems.},
author = {Michał Machura, Saharon Shelah, Boaz Tsaban},
journal = {Fundamenta Mathematicae},
keywords = {pseudointersection number; linear refinement number; forcing; Mathias forcing; $\omega $-cover; $\gamma $-cover; $\tau $ -cover; $\tau $cover; selection principles},
language = {eng},
number = {1},
pages = {15-40},
title = {The linear refinement number and selection theory},
url = {http://eudml.org/doc/286432},
volume = {234},
year = {2016},
}

TY - JOUR
AU - Michał Machura
AU - Saharon Shelah
AU - Boaz Tsaban
TI - The linear refinement number and selection theory
JO - Fundamenta Mathematicae
PY - 2016
VL - 234
IS - 1
SP - 15
EP - 40
AB - The linear refinement number 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 is a variation of . 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 = = in all models where the continuum is at most ℵ₂, and that the cofinality of is uncountable. Using the method of forcing, we show that and are not provably equal to , and rule out several potential bounds on these numbers. Our results solve a number of open problems.
LA - eng
KW - pseudointersection number; linear refinement number; forcing; Mathias forcing; $\omega $-cover; $\gamma $-cover; $\tau $ -cover; $\tau $cover; selection principles
UR - http://eudml.org/doc/286432
ER -