Borel Edge Colorings for Finite Dimensional Groups

Abstract

We study the potential of Borel asymptotic dimension, a tool introduced recently in arXiv:2009.06721, to help produce Borel edge colorings of Schreier graphs generated by Borel group actions. We find that it allows us to recover the classical bound of Vizing in certain cases, and also use it to exactly determine the Borel edge chromatic number for free actions of abelian groups.

Figures (13)

• Figure 1: The key idea in the proof of Theorem \ref{['th:d+1']}. The color $2d+1$ is used (within the region $V_1$) to "swap parity" along the orbits of $\gamma_1$.
• Figure 2: The key idea in the proof of Theorem 2. Now parity along the horizontal orbits is swapped without the use of an additional color. The regions $A$ and $B$ are labeled so that they can be referenced later.
• Figure 3: A visualization of the setup and proof of Lemma \ref{['lem:general']}. The horizontal edges correspond to $\gamma_1$, and the vertical to $\gamma_2$. The edges pictured are those not given the color $4$. $A$ and $B$ are the regions indicated. The brackets show pairs in $P$. (c) shows the configuration of edges mentioned in condition 5 of the lemma statement. That configuration is used to swap parity for the element of $P$ labeled by (a), as shown. (b) labels an element of $P$ on which no parity swap is needed.
• Figure 4: A partial coloring for $d=2$ following the protocol $x_0$ for $e_1$ using the colors 1 and 2 and also following the protocol $x_0$ for $e_2$ using the colors 3 and 4.
• Figure 5: A typical arrangement of the sets in the proof of Lemma \ref{['lem:std']} for $d=2$.

Theorems & Definitions (68)

• Theorem 1
• Theorem 2
• Theorem 3: dim
• Theorem 4: dim
• Theorem 5: CM,dim
• Theorem 6: CM
• Lemma 1
• proof
• Lemma 2
• proof