Meeting infinitely many cells of a partition once

Abstract.

We investigate several versions of a cardinal characteristic $\frak f$ defined by Frankiewicz. Vojtáš showed ${\frak b} \leq{\frak f}$, and Blass showed ${\frak f} \leq \min({\frak d},{\mbox{\rm unif}}({\bf K}))$. We show that all the versions coincide and that ${\frak f}$ is greater than or equal to the splitting number. We prove the consistency of $\max({\frak b},{\frak s}) <{\frak f}$ and of ${\frak f} < \min({\frak d},{\mbox{\rm unif}}({\bf K}))$.