Borel Sets without Perfectly Many Overlapping Translations
Borel Sets without Perfectly Many Overlapping Translations

Summary/Abstract: We study the existence of Borel sets B ⊆ ω 2 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 set B ⊆ ω 2 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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