Borel sets without perfectly many overlapping translations IV
Volume 177 / 2024
Colloquium Mathematicum 177 (2024), 99-126
MSC: Primary 03E35; Secondary 03E15, 03E50
DOI: 10.4064/cm9104-12-2024
Published online: 8 January 2025
Abstract
We show that, consistently, there exists a Borel set $B\subseteq {}^{\omega }2$ admitting a sequence $\langle \eta _\alpha :\alpha \lt \lambda \rangle $ of distinct elements of ${}^{\omega }2$ such that $(\eta _\alpha +B)\cap (\eta _\beta +B)$ is uncountable for all $\alpha ,\beta \lt \lambda $ but with no perfect set $P$ such that $|(\eta +B)\cap (\nu +B)|\geq 6$ for any distinct $\eta ,\nu \in P$. This answers two questions from our previous works.
