# Edge sets on $\omega$ maximal with respect to chromatic number

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 $$\subseteq$$?

$$\mathcal E_n$$ has maximal elements for every $$n\ge1$$.
Let $$V_1,\dots,V_n$$ be pairwise disjoint infinite sets such that $$V_1\cup\cdots\cup V_n=\omega$$ and let $$E=\{e\in[\omega]^\omega:e\not\subseteq V_i\text{ for all }i\in[n]\}$$. Plainly $$\chi(\omega,E)\le n$$.
Let $$c$$ be any proper $$n$$-coloring of $$(\omega,E)$$. For each $$i\in[n]$$ there is some color $$x_i$$ which occurs infinitely often in $$V_i$$, and that color can not occur in $$\omega\setminus V_i$$. Thus the only proper $$n$$-colorings of $$(\omega,E)$$ are the obvious ones, where the color classes are the sets $$V_i$$. This shows that $$\chi(\omega,E)=n$$ and that $$\chi(\omega,E')\gt n$$ if $$E\subsetneqq E'\subseteq[\omega]^\omega$$.