# Can every number be realised as the chromatic number of a countable hypergraph?

If $$H=(V,E)$$ is a hypergraph and $$\kappa$$ is a cardinal,we say a map $$c:V\to\kappa$$ is a coloring if the restriction $$c\restriction_e$$ of $$c$$ to $$e$$ is non-constant whenever $$e\in E$$ and $$|e|>1$$. The smallest cardinal such that there is a coloring from $$V$$ to that cardinal is denoted by $$\chi(H)$$.

By $$[\omega]^\omega$$ we denote the set of infinite subsets of $$\omega$$.

Given $$n\in\omega,n>1$$ is there $$E\subseteq[\omega]^\omega$$ such that $$\chi(\omega,E)=n$$?

Yes. Partition $$\omega$$ into $$n$$ infinite sets $$V_1,\dots,V_n$$. Let $$H=(\omega,E)$$ where $$E=\{e\in[\omega]^\omega:|\{i:e\cap V_i\ne\emptyset\}|\ge2\}$$. Plainly $$\chi(H)\le n$$. Suppose the vertices of $$H$$ are colored with $$m$$ colors, $$m\lt n$$. For each $$i$$ choose an infinite monochromatic set $$W_i\subseteq V_i$$. By the pigeonhole principle, since $$m\lt n$$, for some $$i\ne j$$ the sets $$W_i,W_j$$ have the same color. Then $$W_i\cup W_j\in E$$ is monochromatic, showing that the coloring is not proper.