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

Question

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 $\chi(\omega,E)=n$?

Answer 1

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.