Borel sets without perfectly many overlapping translations

Abstract

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]