# Sh:522

- Shelah, S. (1999).
*Borel sets with large squares*. Fund. Math.,**159**(1), 1–50. arXiv: math/9802134 MR: 1669643 -
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.

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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} }