Combinatorial subspace
An m-dimensional combinatorial subspace is obtained by taking m disjoint subsets [math]\displaystyle{ W_1,\dots,W_m }[/math] of [math]\displaystyle{ [n], }[/math] fixing the values of all coordinates outside these subsets, and taking all points that equal the fixed values outside the [math]\displaystyle{ W_i }[/math] and are constant on each [math]\displaystyle{ W_i. }[/math] Sometimes one adds the restriction that the maximum of each [math]\displaystyle{ W_i }[/math] is less than the minimum of [math]\displaystyle{ W_{i+1} }[/math]. (In particular, the Hales-Jewett theorem can be straightforwardly generalized to yield a monochromatic subspace of this more stronger kind.) In other words, where a combinatorial line has one set of wildcards, an m-dimensional combinatorial subspace has m sets of wildcards
For example, the following strings form a 2-dimensional combinatorial subspace of the strong kind (the wildcard sets are indicated above by a and b)
[math]\displaystyle{ \begin{matrix} 3&3&1&2&1&3&1&1&2&2&1&2&1&&&&3&3&1&2&1&3&1&1&2&2&2&2&2&&&&3&3&1&2&1&3&1&1&2&2&3&2&3\\ 3&3&2&2&2&3&1&2&2&1&2&2&1&&&&3&3&2&2&2&3&1&2&2&2&2&2&2&&&&3&3&2&2&2&3&1&2&2&2&3&2&3\\ 3&3&3&2&3&3&1&3&2&2&1&2&1&&&&3&3&3&2&3&3&1&3&2&2&2&2&2&&&&3&3&3&2&3&3&1&3&2&2&3&2&3\\ \end{matrix} }[/math]
and the following strings form a 2-dimensional combinatorial subspace of the weaker kind
a b a b a b a b a b a b
113113121311321 113213122311321 113313123311321 113113221312321 113213222312321 113313223312321 113113221313321 113213322313321 113313323313321
There is also a natural notion of a combinatorial embedding of [math]\displaystyle{ [3]^m }[/math] into [math]\displaystyle{ [3]^n. }[/math] Given a string [math]\displaystyle{ x\in[3]^n }[/math] and m disjoint subsets [math]\displaystyle{ W_1,\dots,W_m }[/math] of [math]\displaystyle{ [n], }[/math] send [math]\displaystyle{ y\in[3]^m }[/math] to [math]\displaystyle{ x+\sum_iy_iW_i, }[/math] where this denotes the sequence that takes the value [math]\displaystyle{ y_i }[/math] everywhere in the set [math]\displaystyle{ W_i }[/math] and is equal to x everywhere that does not belong to any [math]\displaystyle{ W_i }[/math]. An m-dimensional combinatorial subspace is the image of a combinatorial embedding.
