# Borel sets without perfectly many overlapping translations IV

## TL;DR

This paper demonstrates the consistent existence of a Borel set with a large family of overlapping translations that do not contain a perfect set with uniformly large intersections, answering open questions in descriptive set theory.

## Contribution

It constructs a Borel set with a specific overlapping translation property, providing a negative answer to previously open questions about perfect sets and overlaps.

## Key findings

- Existence of a Borel set with uncountable overlaps without perfect large intersections
- Answers to two open questions in descriptive set theory
- Provides a consistent example under set-theoretic assumptions

## Abstract

We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no perfect set P such that |(eta+B) cap (nu+B)|>5 for any distinct eta,nu from P. This answers two questions from our previous works.