Borel sets without perfectly many overlapping translations

TL;DR

This paper constructs a specific Borel set in the Cantor space using ccc forcing, demonstrating the existence of a large family of overlapping translations without a perfect subset, extending Shelah's earlier work.

Contribution

It introduces a new ccc forcing method to produce Borel sets with large overlapping translations lacking perfect subsets, advancing understanding of translation overlaps in descriptive set theory.

Findings

Existence of Borel sets with large translation overlaps without perfect subsets

Construction of a ccc forcing notion for such sets

Extension of Shelah's previous results on translation overlaps

Abstract

For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134