We say that a [hypergraph](https://en.wikipedia.org/wiki/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?


1. $(V,E\_0)$ is connected, and
2. whenever $E'\subseteq E\_0$ with $E'\neq E\_0$ then $(V,E')$ is no longer connected.



Connected hypergraphs
Let $V=E=\omega$. Connected subgraphs are precisely those which contain as edges $n$ for arbitrarily large $n$. Clearly there is no minimal such subgraph.


