 Extended Hales-Jewett theorem

For every alphabet size $m$, number of colors $k$, and dimension $d$, some $n$ has the property that every $k$-coloring of $[m{]}^n$ contains a monochromatic $d$-dimensional combinatorial subspace. It follows from the Hales-Jewett theorem by identifying $[m{]}^{dN}$ with the word space over the alphabet $[m{]}^d$.