Say you have a graph drawn out in 2D space, but some of the edges cross, and there's no way to arrange the edges so that there are no crossings in 2D. It happens that, by moving the graph into 3D, you may be able to rearrange the edges so that they don't cross.
- By increasing the dimension of the space by $1$, are you guaranteed to have no crossings?
- If not, then how can you go about finding the minimum dimension of space in which the edges don't cross?