While not all graphs can be drawn in R^{2}, every single finite graph can be drawn in the 3 dimensional space R^{3}. The example I will use is called a book embedding.

Imagine you put all of the vertices on the same line in R^{3}. There are an infinite number of planes that go through every point on that line, and do not overlap anywhere else.

You can put each edge on a distinct plane, and they do not overlap, so it is a valid embedding in R^{3}.

In fact, you don’t need to have one plane for each edge. You can put multiple edges on the same plane and they still don’t cross each other.

The minimum number of pages you need to embed a graph is constant no matter which order you put the vertices on the line.