We say that a hypergraph $H=(V,E)$ is *connected* if the following condition holds:

for all $S\subseteq V$ with $\emptyset\neq S \neq V$ there is $e\in E$ that meets both $S$ and $V\setminus S$, i.e. $$S\cap e \neq \emptyset \neq (V\setminus S)\cap e.$$

Given $H=(V,E)$ connected, is there $E_0\subseteq E$ with the following properties?

- $(V,E_0)$ is connected, and
- whenever $E'\subseteq E_0$ with $E'\neq E_0$ then $(V,E')$ is no longer connected.