The linear refinement number is the minimal cardinality of a centered family in such that no linearly ordered set in 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm124-8-2015,
author = {Micha\l\ Machura and Saharon Shelah and Boaz Tsaban},
title = {The linear refinement number and selection theory},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {15-40},
zbl = {06602779},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm124-8-2015}
}
Michał Machura; Saharon Shelah; Boaz Tsaban. The linear refinement number and selection theory. Fundamenta Mathematicae, Tome 233 (2016) pp. 15-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm124-8-2015/