Continuous edge chromatic numbers of abelian group actions

We prove that for any generating set S of Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb Z}^{n}$$\end{document}, the continuous edge chromatic number χ′c(G) of the Schreier graph of the Bernoulli shift action G=F(S,2Zn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=F(S, \ 2^{{\mathbb Z}^{n}})$$\end{document} is χ′(G) + 1 = ∣S∣ + 1. In particular, for the standard generating set, the continuous edge chromatic number of F(2Zn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(2^{{\mathbb Z}^{n}})$$\end{document} is 2n + 1.