If $H=(V,E)$ is a hypergraph, and $\kappa \neq \emptyset$ is a cardinal, a map $c: V\to \kappa$ is said to be a *coloring* if the restriction $c|_e: e\to \kappa$ is non-constant whenever $e\in E$ and $e$ has at least $2$ elements. The least non-empty cardinal such that there is a coloring onto that cardinal is said to be the *chromatic number* of $H$.

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. For $n\in \omega\setminus\{0,1\}$, let $${\cal E}_n =\{E\subseteq [\omega]^\omega: \chi(\omega, E) = n\}.$$

**Question.** For what values of $n$ does ${\cal E}_n$ have maximal elements with respect to $\subseteq$?