The knowledge gap for the chromatic number of (3D) space is even wider than for the plane: it is only known to satisfy 6≤χ(3)≤15. See [Gra04a,Gra04b] for further results and references.
There is now some evidence that the chromatic number of the plane may depend on the axioms of set theory. This was first seen possible in examples constructed by Saharon Shelah and Alexander Soifer. Now Payne [Pay09] has constructed unit-distance graphs with the same property.