Let $S$ be a family of subsets of $\mathbb{N}$. Prove that the following are equivalent: (i) Whenever $N$ is finitely coloured, some member of $S$ is monochromatic. (ii) There is an ultrafilter $U$ such that every set $A \in U$ contains some member of $S$.
The reverse direction is obvious, as each finite colouring of $N$ has one of the colours as a set in $U$, which gives a monochromatic member of $S$. I do not see how to show the reverse direction.
