AI 中文总结
本文证明了 Bernoulli 移位 \(\mathbb Z^2\curvearrowright 2^{\mathbb Z^2}\) 的自由部分上的有向 Schreier 图的连续定向色数为 7。
AI 中文摘要
设 \(\vec F(2^{\mathbb Z^2})\) 为 Bernoulli 移位 \(\mathbb Z^2\curvearrowright 2^{\mathbb Z^2}\) 的自由部分上的有向 Schreier 图，其弧沿两个坐标方向。我们证明它的连续定向色数为 7，即存在一个 7 个顶点的竞赛图，它接收来自 \(\vec F(2^{\mathbb Z^2})\) 的连续图同态，并且不存在从 \(\vec F(2^{\mathbb Z^2})\) 到任何 6 个顶点的竞赛图的连续图同态。
英文摘要
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.
Comments24 pages
