Coarse selectors of graphs

Abstract

We consider a connected graph $\Gamma$ as a coarse space and prove that $\Gamma$ admits a 2-selector if and only if $\Gamma$ is either bounded or coarsely equivalent to $\mathbb{N}$ or $\mathbb{Z}$. We apply this result to geodesic metric spaces admitting linear orders compatible with coarse structures.

coarse structure, 2-selector of a graph, linear order compatible with a coarse structure. arXiv admin note: text overlap with arXiv:2102.03790, arXiv:2102.02053