# Universal sets for ideals

## TL;DR

This paper explores the existence and construction of universal sets for various ideals, demonstrating their minimal Borel complexity and applications in set theory and forcing extensions.

## Contribution

It introduces universal sets for classic and complex ideals, showing their minimal Borel complexity and utility in set-theoretic and forcing contexts.

## Key findings

- Existence of minimal Borel complexity universal sets for null and meager ideals.
- Construction of universal sets for ideals generated by closed null sets and related forcing ideals.
- Universal sets for Fubini products of ideals with specific descriptive set-theoretic complexity.

## 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 $2^\omega$ 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 $\sigma$-ideal generated by closed null subsets of $2^\omega$, and for some ideals connected with forcing notions: $\mathcal{K}_\sigma$ subsets of $\omega^{\omega}$ and the Laver ideal. We also consider Fubini products of ideals and show that there are $\Sigma^0_3$ universal sets for $\mathcal{N}\otimes\mathcal{M}$ and $\mathcal{M}\otimes\mathcal{N}$.