Borel sets with large squares
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 x R contains a musquare (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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 1998
 arXiv:
 arXiv:math/9802134
 Bibcode:
 1998math......2134S
 Keywords:

 Mathematics  Logic