Abstract: Suppose that $X$ is a Hausdorff space, $\mathcal{I}$ is a $\sigma$-ideal on $X$, and $( A_i )_{i \in \omega}$ is a sequence of analytic subsets of $X$. We investigate the circumstances under which there exists $I \in [\omega]^{\omega}$ with $\bigcap_{i \in I} A_i \notin \mathcal{I}$. We focus on Laczkovich-style characterizations and ideals associated with descriptive set-theoretic dichotomies.