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

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

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

Consider for $n\in \mathbb{N}$, $n>1$ the set $[n] = \{1,\ldots,n\}$. We say that a collection ${\cal C}\subseteq {\cal P}([n])$ is *complete linear* if

- $|{\cal C}|=n$,
- $c\neq d \in {\cal C} \implies |c\cap d| = 1$, and
- $|c| > 1$ for all $c\in{\cal C}$.

Given $n>1$, does every complete linear collection on $[n]$ have a chromatic self-map?