Combinatorial subspace

An m-dimensional combinatorial subspace of [math]\displaystyle{ [3]^n }[/math] can be described by a string of length n using the alphabet 1,2,3 together with m additional wildcards, with each wildcard being used at least once. One can then embed [math]\displaystyle{ [3]^m }[/math] into [math]\displaystyle{ [3]^n }[/math] by using the length m string in [math]\displaystyle{ [3]^m }[/math] to fill out the wildcards.

For instance, the string 3y1xy describes a two-dimensional subspace of [math]\displaystyle{ [3]^5 }[/math], given by the grid

33113  33123  33133
32112  32122  32132
31111  31121  31131

A one-dimensional combinatorial subspace is the same thing as a combinatorial line. Note that any combinatorial embedding of [math]\displaystyle{ [3]^m }[/math] in [math]\displaystyle{ [3]^n }[/math] will map combinatorial lines to combinatorial lines.