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Borel sets with large squares. (English) Zbl 0941.03052
It is consistent that for every \(\alpha<\omega_1\) there is a Borel set on the plane containing a square of cardinality \(\aleph_{\alpha+1}\) but not a perfect square (square of a perfect set). If, however, \(\kappa>2^{\aleph_0}\) Cohen reals are added then, in the resulting model, if an analytic set contains a \(\kappa\)-square, then it contains a perfect square. This is connected to some problems on Hanf numbers below \(2^{\aleph_0}\). Under MA the properties turn out to be equivalent. In this technically delicate paper, the case of rectangle containment is also investigated.

MSC:
03E15 Descriptive set theory
03E35 Consistency and independence results
03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
03C55 Set-theoretic model theory
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