Borel sets without perfectly many overlapping translations

Andrzej Ros Lanowski; Saharon Shelah

doi:10.4467/20842589RM.19.001.10649

Borel sets without perfectly many overlapping translations

Andrzej Ros Lanowski, Saharon Shelah

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the existence of Borel sets B ⊆ ^ω2 admitting a sequence hη_α : α < λi of distinct elements of ^ω2 such that ||(η_α + B) ∩(η_β + 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 ℵ_α < c, α < ω₁ and 3 ≤ ι < ω, then there exists a Σ⁰ ₂ set B ⊆ ^ω2 which has ℵ_α many pairwise 2ι-nondisjoint translations but not a perfect set of such translations. Our arguments closely follow Shelah [7, Section 1].

Original language	English
Pages (from-to)	3-43
Number of pages	41
Journal	Reports on Mathematical Logic
Issue number	54
DOIs	https://doi.org/10.4467/20842589RM.19.001.10649
State	Published - 2019

Bibliographical note

Publisher Copyright:
© 2019 Jagiellonian University Press. All rights reserved.

Keywords

Borel sets
Cantor space
Forcing
Non-disjointness rank
Perfect set of overlapping translations

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10.4467/20842589RM.19.001.10649

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abstract = "We study the existence of Borel sets B ⊆ ω2 admitting a sequence hηα : α < λi 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].",

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AB - We study the existence of Borel sets B ⊆ ω2 admitting a sequence hηα : α < λi 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].

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Borel sets without perfectly many overlapping translations

Abstract

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Keywords

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