Enrico Zoli

On the Equivalence Between CH and the Existence of Certain $\mathcal{I}$-Luzin Subsets of $\mathbb{R}$

We extend Rothberger's theorem (on the equivalence between Continuum Hypothesis and the existence of Luzin and Sierpi\'nski-sets having power ${\mathfrak{c}$) and certain paradoxical constructions due to Erd\" os. More precisely, by employing a suitable $\sigma$-ideal associated to the $(\alpha,\beta)$-games introduced by Schmidt, we prove that the Continuum Hypothesis holds if and only if there exist subgroups of $(\mathbb{R},+)$ having power ${\mathfrak{c}$ and intersecting every ``absolutely losing'' (respectively, every meager and null) set in at most countably many points.

Continuum Hypothesis, Schmidt's games, $\mathcal{I}$-Luzin sets, $\sigma$-ideals, vector subspaces of $\mathbb R$ over the rationals.

MSC 2000: 03E15, 03E50, 28A05, 91A05.