Meeting infinitely many cells of a partition once (Q1283122)

Let \({\mathfrak f}\) denote the least cardinality of any collection \(C\) of infinite subsets of \(\omega\) such that for every partition of \(\omega\) into finite sets, there is a member of \(C\) which meets infinitely many cells of the partition in exactly one point. \textit{P. Vojtáš} [Comment. Math. Univ. Carol. 28, 173-183 (1987; Zbl 0651.40001)] showed \({\mathfrak b}\leq {\mathfrak f}\), and A. Blass showed \({\mathfrak f}\leq\min({\mathfrak d},\text{unif} ({\mathbf K}))\). The authors show \({\mathfrak s}\leq{\mathfrak f}\) and prove the consistency of \(\max ({\mathfrak b}, {\mathfrak s})<{\mathfrak f}\) and of \({\mathfrak f}<\min ({\mathfrak d}, \text{unif} ({\mathbf K}))\).