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Aravind and Lee-Elkin (1997) gave a proof of the Bell-Kochen-Specker theoremby showing that it is impossible to color the 60 directions from the center ofa 600-cell to its vertices in a certain way. This paper refines that result byshowing that the 60 directions contain many subsets of 36 and 30 directionsthat cannot be similarly colored, and so provide more economical demonstrationsof the theorem. Further, these subsets are shown to be critical in the sensethat deleting even a single direction from any of them causes the proof tofail. The critical sets of size 36 and 30 are shown to belong to orbits of 200and 240 members, respectively, under the symmetries of the polytope. Acomparison is made between these critical sets and other such sets in fourdimensions, and the significance of these results is discussed.

Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem