Abstract: A countable group $G$ induces an equivalence relation on the Polish space $2^G$. Moreover, it induces a natural graph on the elements of $2^G$: two nodes $x,y$ are connected in $2^G$ iff there is a generator $g\in G$ such that $x=g\cdot y$. We present a recent negative result, that for $G=\mathbb{Z}^2$, if we ignore certain degenerate orbits, it is not possible to three-color the induced graph with a continuous function such that no two adjacent nodes have the same color. This is despite the fact that each individual connected component can be two-colored. Since a continuous four-coloring has been shown to exist by Gao and Jackson, this serves as a computation of the continuous chromatic number of the graph: four. If time permits, we will show how the technique can be used to obtain a variety of negative results for continuous functions, in particular continuous graph homomorphisms.

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