Supersaturated ideals

TL;DR

This paper investigates supersaturated ideals, their preservation under certain forcings, and their independence from ZFC plus the existence of an $\omega_1$-saturated $\sigma$-ideal, contributing to set theory and forcing theory.

Contribution

It introduces the concept of supersaturated ideals, proves their preservation under many ccc forcings, and establishes their independence from ZFC with certain saturation assumptions.

Findings

Many well-known ccc forcings preserve supersaturation.

Existence of supersaturated ideals is independent of ZFC plus $\omega_1$-saturation.

Supersaturated ideals relate to saturation properties in set theory.

Abstract

A $\sigma$-ideal $\cal{I}$ on a set $X$ is supersaturated if for every family $\cal{F}$ of $\cal{I}$-positive sets with $|\cal{F}| < \mathrm{add}(\cal{I})$, there exists a countable set that meets every set in $\cal{F}$. We show that many well-known ccc forcings preserve supersaturation. We also show that the existence of supersaturated ideals is independent of ZFC plus "There exists an $\omega_1$-saturated $\sigma$-ideal".