# Borel sets with large squares

@article{Shelah1998BorelSW,
title={Borel sets with large squares},
author={Saharon Shelah},
journal={arXiv: Logic},
year={1998}
}
• S. Shelah
• Published 1998
• Mathematics
• arXiv: Logic
For a cardinal mu we give a sufficient condition (*)_mu (involving ranks measuring existence of independent sets) for: [(**)_mu] if a Borel set B subseteq R x R contains a mu-square (i.e. a set of the form A x A, |A|= mu) then it contains a 2^{aleph_0}-square and even a perfect square, and also for [(***)_mu] if psi in L_{omega_1, omega} has a model of cardinality mu then it has a model of cardinality continuum generated in a nice, absolute way. Assuming MA + 2^{aleph_0}> mu for… Expand
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Andrzej ROS LANOWSKI and Saharon SHELAH BOREL SETS WITHOUT PERFECTLY MANY OVERLAPPING TRANSLATIONS
• 2019
A b s t r a c t. We study the existence of Borel sets B ⊆ 2 admitting a sequence 〈ηα : α < λ〉 of distinct elements of 2 such that ∣(ηα+B)∩ (ηβ +B) ∣∣ ≥ 6 for all α, β < λ but with no perfect set ofExpand

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