The continuous oriented chromatic number of directed Schreier graphs of ${\mathbb{Z}}^2$-shift actions

Abstract

Let $\vec{F}(2^{{\mathbb{Z}}^2})$ be the directed Schreier graph on the free part of the Bernoulli shift ${\mathbb{Z}}^2\curvearrowright 2^{{\mathbb{Z}}^2}$, with arcs in the two coordinate directions. We prove that the continuous oriented chromatic number of it is 7, that is, there is a tournament on 7 vertices receiving a continuous graph homomorphism from $\vec{F}(2^{{\mathbb{Z}}^2})$ and there is no continuous graph homomorphism from $\vec{F}(2^{{\mathbb{Z}}^2})$ to any tournament on 6 vertices. And we prove that the Borel and measurable oriented chromatic number of directed Schreier graph $\vec{F}(2^{{\mathbb{Z}}^n})$, $n>1$ is 5.

arXiv comment24 pages