Consider the 9-point affine plane $\ F_3\times F_3,\ $ over the 3-element Galois field $\ F_3.\ $ Then $\ n=9,\ $ while the respective $\ m(C)=4 < n-1$.

Here $\ C\ $ is the set of affine lines, i.e. sets described by the linear equations (homogeneous and non-homogenous, just like in an elementary school).

The explicit set of straight lines which include point (0\ 0) is:

$$ \{(0\ 0)\ \ (0\ 1)\ \ (0\ 2)\}$$
$$ \{(0\ 0)\ \ (1\ 0)\ \ (2\ 0)\}$$
$$ \{(0\ 0)\ \ (1\ 1)\ \ (2\ 2)\}$$
$$ \{(0\ 0)\ \ (1\ 2)\ \ (2\ 1)\}$$