**Enrico Zoli**

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

**Abstract:**

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.

**Keywords:**

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.