This is question [Selection problem in a collection of non-empty sets](https://mathoverflow.net/questions/228126/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?


1. $a\in {\cal F} \implies |a|\geq 2$,
2. $a\neq b\in {\cal F} \implies |a\cap b| \leq 1$, and
3. 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 $f(a)\neq f(b)$?



Edge chromatic number of hypergraphs
This is equivalent reformulation of Erdös-Faber-Lovász conjecture, see Wikipedia page about it.


<https://en.m.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture>


