Borel Combinatorics of Schreier Graphs of $\mathbb{Z}$-actions
Su Gao, Yingying Jiang, Tianhao Wang
TL;DR
The paper studies the Borel combinatorics of Schreier graphs arising from $b Z$-actions with finite generating sets, proving that the existence of a Borel equivariant map from the free part $F(2^{b Z})$ to a subshift of finite type $X$ is equivalent to the existence of a continuous equivariant map, and that this decision problem is computable. Leveraging the Two-Tiles framework, it shows decidability of Borel combinatorics in this setting and develops an exponential-time algorithm to compute the Borel chromatic number $ chi(G_S)$ for Schreier graphs $G_S$ determined by a generating set $S$. The authors derive general bounds, provide exact values in several special cases (notably $ chi(S)=3$ or $ chi(S)=n+2$ in certain configurations), and present concrete constructions that realize a range of chromatic numbers. These results illuminate the finite-analytic bridge between Borel and continuous combinatorics for $b Z$-actions and contrast with the $b Z^2$-case, contributing to the broader understanding of locally checkable labeling problems in descriptive set theory and its connections to distributed computing.
Abstract
In this paper we consider the Borel combinatorics of Schreier graphs of $\mathbb{Z}$-actions with arbitrary finite generating sets. We formulate the Borel combinatorics in terms of existence of Borel equivariant maps from $F(2^{\mathbb{Z}})$ to subshifts of finite type. We then show that the Borel combinatorics and the continuous combinatorics coincide, and both are decidable. This is in contrast with the case of $\mathbb{Z}^2$-actions. We then turn to the problem of computing Borel chromatic numbers for such graphs. We give an algorithm for this problem which runs in exponential time. We then prove some bounds for the Borel chromatic numbers and give a formula for the case where the generating set has size 4.