Given an ideal \(I\) and an ultrafilter \(U\), both on \(\omega\), we say that \(U\) is an \(I\)-ultrafilter if for any \(f:\omega\to\omega\) there is \(A\in U\) such that \(f[A]\in I\). This notion was introduced by J. Baumgartner in 1995, and it has proved to be very useful in the classification of combinatorial properties of ultrafilters. In particular, the notion of Hausdorff ultrafilter is codify as being \(\mathcal{G}_{fc}\)-ultrafilter, where \(\mathcal{G}_{fc}\) denotes the ideal of finite chromatic graphs on. We will prove that consistently there is no \(I\)-ultrafilter for any \(F_\sigma\) ideal \(I\). Since the ideal \(\mathcal{G}_{fc}\) is an \(F_\sigma\) ideal, our result implies that consistently there is no Hausdorff ultrafilter. This answers a question of M. Di Nasso and M. Forti, among several other questions about the existence of \(I\)-ultrafilters.