  We show that, for every linear ordering of $[2]^n$, there is a large subcube
on which the ordering is lexicographic. We use this to deduce that every long
sequence contains a long monotone subsequence supported on an affine cube.
  More generally, we prove an analogous result for linear orderings of $[k]^n$.
We show that, for every such ordering, there is a large subcube on which the
ordering agrees with one of approximately $\frac{(k-1)!}{2(\ln 2)^k}$
orderings.
