Borel sets without perfectly many overlapping translations  

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]

Received 16 June 2018

Publication 1138 of the second author.

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

[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. 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.