an:01295193
Zbl 0941.03052
Shelah, Saharon
Borel sets with large squares
EN
Fundam. Math. 159, No. 1, 1-50 (1999).
00056624
1999
j
03E15 03E35 03E05 03E50 03C55
Borel set; perfect square; Cohen reals; analytic set; Hanf numbers; rectangle containment; Martin's Axiom; Borel rectangles
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.
Peter Komjath (Budapest)