Failure of the $G_0$-dichotomy for generalized Cantor spaces

Kechris, Solecki and Todorčević’s $G_0$-dichotomy characterizes Borel graphs that admit a Borel measurable coloring with countably many colors. We show that the analogue to the $G_0$-dichotomy for generalized Cantor spaces fails at the lowest possible complexity, namely for closed graphs.