A linear hypergraph is a hypergraph $H=(V,E)$ such that
- for $e\in E$ we have $|e|\geq 2$, and
- if $e\neq e_1\in E$, then $|e\cap e_1| \leq 1$.
An injective edge choice function of a linear hypergraph is an injective map $f:E\to V$ such that:
for all $e\in E$ we have $f(e)\in e$.
Obviously, if $|E|>|V|$ there cannot be such a function.
Question. If $H=(V,E)$ is a linear hypergraph with $V$ finite and $|e|\geq 3$ for all $e\in E$, does $H$ necessarily have an injective edge choice function?