Borel sets without perfectly many overlapping translations

• Autorzy: Rosłanowski Andrzej , Shelah Saharon

Identyfikatory

DOI

10.4467/20842589RM.19.001.10649

• Języki publikacji: EN

Abstrakty

EN

We study the existence of Borel sets B ⊆ ω2 admitting a sequence ηα : α<λ 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].

Słowa kluczowe

EN

forcing; Borel sets; Cantor space; perfect set of overlapping translations; non-disjointness rank

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2019
• Tom: Vol. 54
• Strony: 3--43
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Rosłanowski Andrzej

• Department of Mathematics University of Nebraska at Omaha Omaha, NE 68182-0243, USA

autor

Shelah Saharon

• Institute of Mathematics The Hebrew University of Jerusalem 91904 Jerusalem, Israel
• Department of Mathematics Rutgers University New Brunswick, NJ 08854, USA, http://shelah.logic.at

Bibliografia

• [1] M. Balcerzak, A. Rosłanowski, and S. Shelah, Ideals without ccc, Journal of Symbolic Logic 63 (1998), 128–147, arxiv:math/9610219.
• [2] T. Bartoszyński 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. Roslanowski 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. Roslanowski, 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 σ–ideal on 2N, Proc. Amer. Math. Soc. 141 (2013), 1055–1065.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).