Supersaturated ideals

Abstract

A $\ddot{I}ƒ$-ideal $\mathcal{I}$ on a set $X$ is supersaturated if for every family $\mathcal{F}$ of $\mathcal{I}$-positive sets with $|\mathcal{F}|<\mathrm{add}(\mathcal{I})$, there exists a countable set that meets every set in $\mathcal{F}$. 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 $\ddot{I}{‰}_1$-saturated $\ddot{I}ƒ$-ideal".

11 pages, submitted