### 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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