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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 $\chi(\omega, {\cal A}) = n$?

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    $\begingroup$ Maybe I am misunderstanding the question, but isn't this already true for graphs (which are hypergraphs with pairwise almost disjoint sets). $\endgroup$
    – 1001
    Commented Apr 9 at 21:10

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This question was answered affirmatively by Theorem 1.1 of Paul Erdős and Saharon Shelah, Separability properties of almost-disjoint families of sets, Israel J. Math. 12 (1972) 207–214 (pdf), improving earlier results of S. Hechler, Classifying almost-disjoint families with applications to $\beta N\setminus N$, Israel J. Math. 10 (1971) 413–432.

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