Borel sets without perfectly many overlapping translations, II

TL;DR

This paper constructs a specific Sigma^0_2 subset of the Cantor space with unique translation intersection properties, demonstrating limitations on the existence of perfect sets of translations in certain models.

Contribution

It introduces a novel construction of a Sigma^0_2 set with controlled translation intersections, resolving a previously open problem about perfect sets of translations.

Findings

Constructed a Sigma^0_2 set with specific intersection properties

Showed such sets lack perfect sets of translations in ccc extensions

Established the existence of uncountably many translations with certain intersection sizes in ZFC

Abstract

For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor space for which one may force aleph_epsilon translations with intersections of size 2i, but such that it has no perfect set of such translations in any ccc extension. These sets have uncountably many translations with intersections of size 2i in ZFC, so this answers Problem 3.4 of arxiv:1711.04058 .