  We investigate maximal free sequences in the Boolean algebra
$\mathcal{P}(\omega)/\mathrm{fin}$, as defined by D. Monk. We provide some
information on the general structure of these objects and we are particularly
interested in the minimal cardinality of a free sequence, a cardinal
characteristic of the continuum denoted $\mathfrak{f}$. Answering a question of
Monk, we demonstrate the consistency of $\omega_1 = \mathfrak{i} = \mathfrak{f}
< \mathfrak{u} = \omega_2$. In fact, this consistency is demonstrated in the
model of S. Shelah for $\mathfrak{i} < \mathfrak{u}$. Our paper provides a
streamlined and mostly self contained presentation of this construction.
