Let $l\geq 3$ be an integer. Is there $n\in\mathbb{N}$ and a hypergraph $H=(\{1,\ldots,n\},E)$ with the following properties?

- for all $e\in E$ we have $|e| \geq l$
- $e_1\neq e_2 \in E \implies |e_1 \cap e_2|\leq 1 $
- every edge intersects at least $n$ other edges, that is, for every $e\in E$ we have $|\{d\in E: d\neq e \text{ and } d\cap e\neq \emptyset\}|\geq n.$

(It would also be nice to know what the smallest value for $n$ is, given any $l \geq 3$. Obviously $n\geq 2l -1$ but this is a very weak lower bound.)