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 $f(a)\neq f(b)$?