Abstract
We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.
Highlights
There is some interest in the literature in Borel sets admitting many pairwise disjoint translations
Balcerzak, Roslanowski, and Shelah [1] studied the σ–ideal of subsets of ω2 generated by Borel sets with a perfect set of pairwise disjoint translations
We show that it is consistent with ZFC that there is a Σ02 subset B of the Cantor space ω2 such that
Summary
There is some interest in the literature in Borel sets admitting many pairwise disjoint translations. Balcerzak, Roslanowski, and Shelah [1] studied the σ–ideal of subsets of ω2 generated by Borel sets with a perfect set of pairwise disjoint translations. Every uncountable Borel subset of ω2 has a perfect set of pairwise non-disjoint translations. We fully utilize the algebraic properties of (ω2, +), in particular the fact that all elements of ω2 are self-inverse. This line of research will be continued in Roslanowski and Shelah [4], where we will deal with the general case of κ many pairwise non-disjoint translations (getting the full parallel of [5, Theorem 1.13]). For a forcing notion P, all P–names for objects in the extension via P will be denoted with a for the generic tilde filter below in P
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