If $H=(V,E)$ is a hypergraph then the chromatic number $\chi(H)$ is defined to be the least cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ with $|e| \geq 2$ the restriction $c\restriction_e: e\to \kappa$ is not constant.
We say that $A, B\subseteq \omega$ are almost disjoint if $|A\cap B|$ is finite.
Given $n\in (\omega\setminus\{0,1\})\cup \{\omega\}$ is there a collection ${\cal A}\subseteq {\cal P}(\omega)$ of pairwise almost disjoint infinite sets such that $\chi(\omega, {\cal A}) = n$?