Borel sets with large squares

Abstract

This is a slightly corrected version of an old work. For a cardinal $\mu$ we give a sufficient condition ${\oplus }_{\mu }$ (involving ranks measuring existence of independent sets) for: ${\otimes }_{\mu }$ if a Borel set $B\subseteq \mathbb{R}\times \mathbb{R}$ contains a $\mu$-square (i.e. a set of the form $A\times A$, with $|A|=\mu )$ then it contains a $2^{{\aleph }_0}$-square and even a perfect square. And also for ${\otimes }_{\mu }^{\prime }$ 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 $\mathrm{MA}+2^{{\aleph }_0}>\mu$ for transparency, those three conditions (${\oplus }_{\mu },{\otimes }_{\mu }$ and ${\otimes }^{\prime }{e}_{\mu }$) are equivalent, and by this we get e.g. $\underset{\alpha <{\omega }_1}{\bigwedge }[2^{{\aleph }_0}\ge {\aleph }_{\alpha }\Rightarrow \neg {\otimes }_{{\aleph }_{\alpha }}$], and also $\min \{\mu :{\otimes }_{\mu }\}$ has cofinality ${\aleph }_1$ if it is $<2^{{\aleph }_0}$. We deal also with Borel rectangles and related model theoretic problems.

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