# Connected hypergraphs

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?

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.

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.
• Could you clarify on the edges? Is there only one edge, or every edge is an element of $ω$? Commented Sep 18, 2019 at 11:29
• @Bullet51 $E=\omega=\{0,1,2,3,\dots\}=\{\{\},\{0\},\{0,1\},\dots\}$. That is, edges are the (proper) initial segments of $\omega$. Perhaps I went a little too far trying to make my answer as concise as possible :P Commented Sep 18, 2019 at 11:32