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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.

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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