Small uncountable cardinals in large-scale topology

Abstract

In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete, inseparated, large) of finitary or locally finite coarse structures on $\omega$. Besides well-known cardinals $\mathfrak b,\mathfrak d,\mathfrak c$ we shall encounter two new cardinals $\mathsf \Delta$ and $\mathsf \Sigma$, defined as the smallest weight of a finitary coarse structure on $\omega$ which contains no discrete subspaces and no asymptotically separated sets, respectively. We prove that $\max\{\mathfrak b,\mathfrak s,\mathrm{cov}(\mathcal N)\}\le\mathsf \Delta\le\mathsf \Sigma\le\mathrm{non}(\mathcal M)$, but we do not know if the cardinals $\mathsf \Delta,\mathsf \Sigma,\mathrm{non}(\mathcal M)$ can be separated in suitable models of ZFC.