Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say that ${\cal A}\subseteq [\omega]^\omega$ is an *almost disjoint family* if $A \neq B \in {\cal A}$ implies $|A\cap B|< \aleph_0$.

Let $X\neq\varnothing$ be a set and let ${\cal E}\subseteq {\cal P}(X)\setminus\{\varnothing\}$ be a collection of non-empty subset. We say that a map $f: {\cal E}\to X$ is a *chromatic self-map* if



1. $f(e) \in e$ for all $e\in {\cal E}$, and



2. if $e_1\neq e_2 \in {\cal E}$ and $e_1\cap e_2 \neq \varnothing$, then $f(e_1)\neq f(e_2)$.

**Question.** Does every almost disjoint family ${\cal A}\subseteq [\omega]^\omega$ have a chromatic self-map?

**Remark.** It suffices to answer the question for maximum almost disjoint families ("MAD families").