# Sh:522

• Shelah, S. (1999). Borel sets with large squares. Fund. Math., 159(1), 1–50.
• Abstract:
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\times R contains a \mu-square (i.e. a set of the form A \times 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 transparency, those three conditions ((*)_\mu, (**)_\mu and (***)_\mu) are equivalent, and by this we get e.g. (\forall\alpha<\omega_1)(2^{\aleph_0}\geq \aleph_\alpha \Rightarrow \neg (**)_{\aleph_\alpha}), and also \min\{\mu:(*)_\mu\}, if <2^{\aleph_0}, has cofinality \aleph_1. We deal also with Borel rectangles and related model theoretic problems.

Bib entry
@article{Sh:522,
author = {Shelah, Saharon},
title = {{Borel sets with large squares}},
journal = {Fund. Math.},
fjournal = {Fundamenta Mathematicae},
volume = {159},
number = {1},
year = {1999},
pages = {1--50},
issn = {0016-2736},
mrnumber = {1669643},
mrclass = {03E05 (03C55 03E15 03E35 03E50)},
note = {\href{https://arxiv.org/abs/math/9802134}{arXiv: math/9802134}},
arxiv_number = {math/9802134}
}