## Graphs with minimum degree $\delta(G)\lt\aleph_0$

Let $$G=(V,E)$$ be a graph with minimum degree $$\delta(G)=n\lt\aleph_0$$. Does $$G$$ necessarily have a spanning subgraph $$G’=(V,E’)$$ which also has minimum degree $$\delta(G’)=n$$ and is minimal with that property?

This question is easily answered in the affirmative if $$G$$ is locally finite or if $$n\le1$$. It already seems difficult for $$n=2$$, but I am not very clever and may be missing something obvious.

The question also seems to make sense for hypergraphs:

Let $$m,n\in\mathbb N$$. Let $$E$$ be a family of sets, each of cardinality at most $$m$$. If $$E$$ is an $$n$$-cover of a set $$V$$ (each element of $$V$$ is in at least $$n$$ elements of $$E$$), does $$E$$ contain a minimal $$n$$-cover of $$V$$?

I would expect such simple questions to have been asked and answered 100 years ago.

Where are these questions considered in the literature?