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].

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Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).