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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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