This is question Selection problem in a collection of non-empty sets with a simplification in criterion 3.

Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties?

- $a\in {\cal F} \implies |a|\geq 2$,
- $a\neq b\in {\cal F} \implies |a\cap b| \leq 1$, and
there is no function $f: {\cal F} \to X$ such that

- if $a, b\in {\cal F}$ with $a\ne b$ and $a\cap b\neq \emptyset$ then $f(a)\neq f(b)$?

Selectionproblem any more since $f(a)\in a$ is not required. I'd call it "edge chromatic numbers of hypergraphs". Of course there is no such $\mathcal F$ with $X$ infinite or with $|a|=2$ for all $a\in\mathcal F.$ A projective plane won't do because the number of lines is equal to the number of points. $\endgroup$ – bof Jan 11 '16 at 23:48