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.