If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $V\neq\varnothing\neq E$ and $|e| > 1$ for all $e\in E$, and $\kappa\neq\varnothing$ is a cardinal, we say that a map $c:V\to\kappa$ is a *coloring* if the restriction $c\restriction\_e: e\to \kappa$ is non-constant for each $e\in E$. We denote by $\chi(H)$ the smallest cardinal $\kappa$ such that there is a coloring $c:V \to \kappa$.


Assume the Axiom of Choice. If ${\cal A, B}$ are infinite [maximal almost disjoint families](https://en.wikipedia.org/wiki/Almost_disjoint_sets) on $\omega$, do we necessarily have $\chi((\omega, {\cal A}))=\chi((\omega,{\cal B}))$?



Does every maximal almost disjoint family have the same chromatic number?
A negative answer to the question follows by the proof of Theorem 1.1 in the following paper of Erdős and Shelah, where for every $n<\omega$ they construct a mad family that is $(n+1)$-colorable but not $n$-colorable:


*Erdős, Paul; Shelah, Saharon*, [**Separability properties of almost-disjoint families of sets**](http://dx.doi.org/10.1007/BF02764666), Isr. J. Math. 12, 207-214 (1972). [ZBL0246.05002](https://zbmath.org/?q=an:0246.05002).


It should also be noted that a completely separable mad family can't have a finite chromatic number: given any finite coloring, there is a color $n$ such that the set $x$ of integers getting that color belongs to the coideal generated by the mad family. By complete separability, $x$ contains a monochromatic member of the family. I won't be surprised if there are also $ZFC$ constructions of such families, but I haven't thought about it enough.


