Infinite graphs can be embedded in $\mathbb{R}^3$

Question

It is well known that any finite graph can be embedded into a three-dimensional space, but what happens to infinite graphs? I know that for example, the order-7 triangular tiling has a hyperbolic geometry, and I assume it can be embedded into $\mathbb{H}^2$ (and hence in $\mathbb{R}^{3}$ (?)). But I have no idea if there are infinite graphs that need more dimensions. Can you give me some help on this?

Answer 1

If the cardinality is larger than the cardinality of $\mathbb R$ then it is not possible.

For other graphs I think it is possible.

It suffices to do it for a complete graph with vertex set of cardinality $\mathbb R$. We do this by letting our vertices be the points on the $z$ axis.

Then we can assign a unique plane containing the $z$ axis to every pair of vertices (because $\mathbb R^2$ has the same cardinality as $\mathbb R$.

Finally we can make the path between two distinct vertices be an arc that is contained in that plane (and only touches the $z$-axis at the end points).