Combinatorial subspace

An m-dimensional combinatorial subspace is obtained by taking m disjoint subsets $W_1,\dots,W_m$ of $[n],$ fixing the values of all coordinates outside these subsets, and taking all points that equal the fixed values outside the $W_i$ and are constant on each $W_i.$ Sometimes one adds the restriction that the maximum of each $W_i$ is less than the minimum of $W_{i+1}$. (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 $[3]^m$ into $[3]^n.$ Given a string $x\in[3]^n$ and m disjoint subsets $W_1,\dots,W_m$ of $[n],$ send $y\in[3]^m$ to $x+y_1W_1+\dots+y_mW_m,$ where this denotes the sequence that takes the value $y_i$ everywhere in the set $W_i$ and is equal to x everywhere that does not belong to any $W_i$. An m-dimensional combinatorial subspace is the image of a combinatorial embedding.