Borel sets without perfectly many overlapping translations  

Andrzej Rosłanowski,

Saharon Shelah

Abstrakt

We study the existence of Borel sets B  2 admitting a sequence  :  <  of distinct elements of 2 such that (+B)( +B)  6 for all ,  <  but with no perfect set of such ’s. Our result implies that under the Martin Axiom, if  < c,  < 1 and 3   < , then there exists a  02 set B  2 which has  many pairwise 2–nondisjoint translations but not a perfect set of such translations. Our arguments closely follow Shelah [7, Section 1].

AMS subject classification: Primary 03E35; Secondary 03E15, 03E50.

 

Słowa kluczowe: Forcing, Borel sets, Cantor space, perfect set of overlapping translations, non-disjointness rank
References

[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. Ros–łanowski 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. Rosłlanowski, 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.

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