A *saturated linear hypergraph* is a hypergraph $H=(V,E)$ such that

- $|e|\geq 2$ for all $e\in E$,
- $|e_1\cap e_2| = 1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$, and
- $|\{e\in E:v\in e\}| = 2.$

Let $E$ be the set of $n\in\mathbb{N}$ such that it is impossible to have a saturated linear hypergraph on $\{1,\ldots, n\}$. (For instance, $4\in E$.) Is $E$ infinite?