Bases and Borel selectors for tall families

0503652 - MÚ 2020 RIV US eng J - Journal Article
Grebík, Jan - Uzcátegui, C.
Bases and Borel selectors for tall families.
Journal of Symbolic Logic. Roč. 84, č. 1 (2019), s. 359-375. ISSN 0022-4812. E-ISSN 1943-5886
R&D Projects: GA ČR GF15-34700L
Institutional support: RVO:67985840
Keywords : Borel ideals * Borel selector * Galvin's Lemma * tall families
OECD category: Pure mathematics
Impact factor: 0.642, year: 2019
Method of publishing: Open access
http://dx.doi.org/10.1017/jsl.2018.66

Given a family of infinite subsets of N, we study when there is a Borel function S: 2N → 2 N such that for every infinite x insin, 2 N , S(X) ⊂ X and. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams' theorem admits such a Borel selector. However, we also show that the analogous result for Galvin's lemma is not true by proving that there is an Fσ tall ideal on N without a Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a pi, 1 2 tall ideal on without a tall closed subset.
Permanent Link: http://hdl.handle.net/11104/0295463