Solution
= Solution
Assume condition ii. In a finite coloring
$$
\mathbb N=C_1\sqcup\cdots\sqcup C_r,
$$
an <ultrafilter> contains exactly one color class: at least one must belong to it because their union is $\mathbb N$, and two disjoint classes cannot both belong to a proper filter. The chosen $C_i$ contains a member of $\mathcal S$, which is therefore monochromatic. This proves condition i.