Borel sets without perfectly many overlapping translations
Choose format
RIS BIB ENDNOTEBorel sets without perfectly many overlapping translations
Publication date: 08.10.2019
Reports on Mathematical Logic, 2019, Number 54, pp. 3-43
https://doi.org/10.4467/20842589RM.19.001.10649Authors
Borel sets without perfectly many overlapping translations
We study the existence of Borel sets B ⊆ ω2 admitting a sequence 〈ηα : αλ〉 of distinct elements of ω2 such that |(ηα +B)∩(ηβ +B)| ≥ 6 for all α,β λ but with no perfect set of such η’s. Our result implies that under the Martin Axiom, if ℵα c, α ω1 and 3 ≤ ι ω, then there exists a Σ0 2 set B ⊆ ω2 which has ℵα many pairwise 2ι–nondisjoint translations but not a perfect set of such translations. Our arguments closely follow Shelah [7, Section 1]
[1] M. Balcerzak, A. Rosłanowski, and S. Shelah, Ideals without ccc, Journal of Symbolic Logic 63 (1998), 128–147, arxiv:math/9610219.
[2] T. Bartoszy´nski and H. Judah, Set Theory: On the Structure of the Real Line, A.K. Peters, Wellesley, Massachusetts, 1995.
[3] M. Elekes and T. Keleti, Decomposing the real line into Borel sets closed under addition, MLQ Math. Log. Q. 61 (2015), 466–473.
[4] T. Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, the third millennium edition, revised and expanded.
[5] A. Rosłanowski and V.V. Rykov, Not so many non-disjoint translations, Proceedings of the American Mathematical Society, Series B 5 (2018), 73–84, arxiv:1711.04058.
[6] A. Rosłanowski, and S. Shelah, Borel sets without perfectly many overlapping translations II. In preparation.
[7] S. Shelah, Borel sets with large squares, Fundamenta Mathematicae 159 (1999), 1–50, arxiv:math/9802134.
[8] P. Zakrzewski, On Borel sets belonging to every invariant ccc –ideal on 2N, Proc. Amer. Math. Soc. 141 (2013), 1055–1065.
Information: Reports on Mathematical Logic, 2019, Number 54, pp. 3-43
Article type: Original article
Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-0243, USA
Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA
Published at: 08.10.2019
Received at: 16.06.2018
Article status: Open
Licence: CC BY-NC-ND
Percentage share of authors:
Classification number:
Article corrections:
-Publication languages:
EnglishView count: 1667
Number of downloads: 1121