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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.
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1 Answer 1

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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.

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  • $\begingroup$ Could you clarify on the edges? Is there only one edge, or every edge is an element of $ω$? $\endgroup$ Commented Sep 18, 2019 at 11:29
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    $\begingroup$ @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 $\endgroup$
    – Wojowu
    Commented Sep 18, 2019 at 11:32

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