Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$.
Let $S\subseteq [\omega]^\omega$. We say that a map $c:\omega \to \{0,\ldots,n-1\}$ is a coloring for $S$ with $n$ colors if for all $s\in S$ the restriction $c|_s$ of $c$ to $s$ is non-constant.
What is an example of a set $S\subseteq [\omega]^\omega$ such that $S$ has a coloring with $3$ colors, but not with $2$ colors?
Partition $\omega$ into three infinite subsets $A_0,A_1,A_2$. Let $S$ consists of subsets which intersects precisely two of the $A_i$ at infinitely many elements. It can obviously be $3$-colored. Suppose there was a $2$-coloring, with color classes $c_0,c_1$. Then either $c_0$ or $c_1$ contains infinitely many elements of some two of $A_0,A_1,A_2$, say $c_0\cap A_0,c_0\cap A_1$ are infinite. Then $c_0\cap(A_0\cup A_1)\in S$ is monochromatic.
This can be generalized to a hypergraph $S\subseteq[\omega]^\omega$ with chromatic number $n$: partition $\omega$ into $A_0,\dots,A_{n-1}$ and let $S$ consists of subsets which intersect two of those at infinitely many elements.
