A *linear hypergraph* is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ such that 


1. for $e\in E$ we have $|e|\geq 2$, and
2. 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?



Injective edge choice functions in linear hypergraphs
Even $|E|\leq|V|$ does not hold: An [affine plane](https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)#Finite_affine_planes) contains more lines than points.


