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

3312131122121   3312131122222   3312131122323  
3322231222121   3322231222222   3322231222323
3332331322121   3332331322222   3332331322323

and the following strings form a 2-dimensional combinatorial subspace of the weaker kind

113113121311321   113213122311321   113313123311321
113113221312321   113213222312321   113313223312321
113113321313321   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+y_1W_1+\dots+y_mW_m, }[/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.