Continuous Edge Chromatic Numbers of Abelian Group Actions

We prove that for any generating set $S$ of ${\mathbb{Z}}^n$, the continuous edge chromatic number of the Schreier graph of the Bernoulli shift action $G=F(S,2^{{\mathbb{Z}}^n})$ is ${\chi }_c^{\mathrm{\prime }}(G)={\chi }^{\mathrm{\prime }}(G)+1$. In particular, for the standard generating set, the continuous edge chromatic number of $F(2^{{\mathbb{Z}}^n})$ is $2n+1$.