Universal sets for ideals

Abstract

In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of $2^ω$ and meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for $\mathcal{E}$ - the $σ$-ideal generated by closed null subsets of $2^ω$, and for some ideals connected with forcing notions: $\mathcal{K}_σ$ subsets of $ω^ω$ and the Laver ideal. We also consider Fubini products of ideals and show that there are $Σ^0_3$ universal sets for $\mathcal{N}\otimes\mathcal{M}$ and $\mathcal{M}\otimes\mathcal{N}$.