Any linear ordering of [2]^n has a large subcube that is lexicographic; generalization bounds the number of possible orderings on subcubes of [k]^n by roughly (k-1)! / (2 (ln 2)^k).
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Linear orderings of combinatorial cubes
Any linear ordering of [2]^n has a large subcube that is lexicographic; generalization bounds the number of possible orderings on subcubes of [k]^n by roughly (k-1)! / (2 (ln 2)^k).