Many simple combinatorial questions have corresponding definable (i.e., continuous, measurable, Borel, etc.) analogues with different answers. For example, it is well known that if a topological space is given a graph structure in a Borel way, there may be graph colorings (e.g., with Choice) that use less colors than any definable coloring, therefore the "definable" chromatic number differs from the classical chromatic number. For the space $F(2^{\mathbb{Z}^2})$ (a variation on the Cantor space), we provide a necessary and sufficient condition for the existence of a continuous equivariant map to a given subshift of finite type; these existence questions constitute a large and varied range of combinatorial questions including chromatic number. Specifically, we show there is a sequence $(X_{n})_{n∈\mathbb{N}}$ of subshifts of finite type such that for any subshift of finite type $X$, there exists a continuous equivariant map $\varphi : F(2^{\mathbb{Z}^{2}}) \rightarrow X$ iff there is a continuous equivariant map $\varphi_{n} : X_{n} \rightarrow X$ for some $n$. The latter is conceptually much simpler; indeed, a finite search for each $n$ will determine the existence of $\varphi_{n}$.