For any cardinal $\alpha \in \omega\cup \{\omega\}$, let $[\omega]^\alpha$ denote the collection of subsets of $\omega$ having cardinality $\alpha$.

A *linear hypergraph* $H=(V,E)$ is a hypergraph such that whenever $e\neq e_1\in E$ we have $|e\cap e_1|\leq 1$.

A *coloring* of a hypergraph $H=(V,E)$ is a map $c:V \to \alpha$, where $\alpha \neq \varnothing$ is a cardinal, such that for all $e\in E$ with $|e|>1$ we have that the restriction $c{\restriction}_e$ is non-constant. We denote by $\chi(H)$ the smallest cardinal such that there is a coloring from $V$ to that cardinal.

If $\alpha \in (\omega\cup\{\omega\})\setminus \{0,1,2\}$, is there a linear hypergraph $H = (\omega, E)$ with $E\subseteq [\omega]^\alpha$ and $\chi(H)=\aleph_0$?