By way of clearing some space off my whiteboard, here's a drawing that's been taking up space in the middle of it for too long:
The context is a problem of Norine on edge-antipodal colorings of cubes: is it the case that, whenever the edges of a hypercube are two-colored in such a way that edges directly opposite each other have opposite colors, then some pair of opposite vertices are connected by a path of edges of only one color?
I was looking at a stronger variation of the problem: if a cube's edges are two-colored arbitrarily, is there always either a monochromatic antipodal path or a monochromatic zone (a family of parallel edges, all given the same color)? If true this would answer Norine's problem, because an antipodally colored cube can't have a monochromatic zone. But it's not true, and the drawing shows why not. It's a six-dimensional hypercube (not antipodally colored), in which there are no monochromatic zones (each class of parallel edges has both colors). But there are also no monochromatic antipodal paths: an antipodal path would have to project to an antipodal path in a little three-dimensional cube as well as one in the big three-dimensional cube, but in the little cube the only antipodal paths are brown and in the big one they are only purple.
There's another variation that I've also thought about, without much success: given an arbitrily two-edge-colored hypercube, must there always exist a path between two opposite vertices in which there exists at most one pair of differently-colored consecutive edges? Norine's problem and this problem don't require the paths to be shortest paths but that would also be interesting.