Borel sets with large squares

by Shelah. [Sh:522]
Fundamenta Math, 1999
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 transparency, those three conditions ((*)_mu, (**)_mu and (***)_mu) are equivalent, and by this we get e.g. (for all alpha < omega_1)(2^{aleph_0} >= aleph_alpha => not (**)_{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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