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



Selection problem in a collection of non-empty sets
Satisfying the new Condition 1, If $|X|\geqslant4$, let ${\cal F}$ consist of all subsets of $X$ with two elements.


Condition 1 can be weakened. For example, if $X=\{a\_{i,j}:1\leqslant i,j\leqslant 3\}$, then the following 12 subsets, each of size 3, satisfy the conditions.
$$
\{a\_{1,1},a\_{1,2},a\_{1,3}\},\{a\_{2,1},a\_{2,2},a\_{2,3}\},\{a\_{3,1},a\_{3,2},a\_{3,3}\}
$$
$$\{a\_{1,1},a\_{2,1},a\_{3,1}\},\{a\_{1,2},a\_{2,2},a\_{3,2}\},\{a\_{1,3},a\_{2,3},a\_{3,3}\}
$$
$$
\{a\_{1,1},a\_{2,2},a\_{3,3}\},\{a\_{1,3},a\_{2,2},a\_{3,1}\}
$$
$$
\{a\_{1,1},a\_{2,3},a\_{3,2}\},\{a\_{1,3},a\_{2,1},a\_{3,2}\},\{a\_{1,2},a\_{2,1},a\_{3,3}\},\{a\_{1,2},a\_{2,3},a\_{3,1}\}
$$


It seems likely that we can permit any minimum size for sets in ${\cal F}$. 


