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A linear hypergraph is a 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?

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  • $\begingroup$ Maybe I misread something, but does 2. hold if $e\cap e_1=\emptyset$ anyway? So the question is if $H$ has a set of distinct representatives.Surely it does not, if $|E|>|V|$. $\endgroup$ – Péter Komjáth May 15 at 15:26
  • $\begingroup$ You are right @PéterKomjáth! I should formulate this in a more elegant way. - An interesting subquestion is whether $|e|>2$ for all $e\in E$ implies $|E|\leq |V|$? $\endgroup$ – Dominic van der Zypen May 15 at 17:52
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    $\begingroup$ No, there's an affine plane on 9 points with 12 lines, and every line has length 3. $\endgroup$ – Bullet51 May 16 at 2:51
  • $\begingroup$ Thanks - can you post this as an answer such that we can close this thread? $\endgroup$ – Dominic van der Zypen May 16 at 7:34
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Even $|E|\leq|V|$ does not hold: An affine plane contains more lines than points.

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