Combinatorial subspace

A $d$-dimensional combinatorial subspace, also called a $d$-parameter set, has $d$ pairwise disjoint nonempty active coordinate sets. Coordinates in the $j$th active set all receive one freely chosen alphabet letter, independently for each $j$, while the remaining coordinates stay fixed.

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 $[m{]}^d$.