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'''Graham-Rothschild theorem''' (k=3), Version 1: If all the [[combinatorial line]]s in <math>[3]^n</math> is partitioned into c color classes, and n is sufficiently large depending on c, m, then there is an m-dimensional [[combinatorial subspace]] of <math>[3]^n</math> such that all the combinatorial lines in this subspace have the same color.

This theorem implies the [[Hales-Jewett theorem]].  It has an alternate formulation:

'''Graham-Rothschild theorem''' (k=3), Version 2: If <math>[4]^n</math> is partitioned into c color classes, and n is sufficiently large depending on c, m, then there is an m-dimensional [[combinatorial subspace]] of <math>[4]^n</math>, with none of the fixed positions equal to 4, such that all elements of this subspace that contain at least one 4 have the same color.

Indeed, one can think of a combinatorial line in <math>[3]^n</math> as a string in <math>[4]^n</math> containing at least one 4, by thinking of 4 as the "wildcard".  It is necessary to restrict to elements containing at least one 4; consider the coloring that colors a string black if it contains at least one 4, and white otherwise.

The Graham-Rothschild theorem and the [[Carlson-Simpson theorem]] have a common generalisation, the [[Carlson-Simpson Graham-Rothschild theorem]].