Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 130 / 4 / i / Solution

Assume condition ii. In a finite coloringan 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.