On Ramsey-type properties of the distance in nonseparable spheres

Abstract

Given an uncountable subset $\mathcal Y$ of a nonseparable Banach space, is there an uncountable $\mathcal Z\subseteq \mathcal Y$ such that the distances between any two distinct points of $\mathcal Z$ are more or less the same? If an uncountable subset $\mathcal Y$ of a nonseparable Banach space does not admit an uncountable $\mathcal Z\subseteq \mathcal Y$, where any two points are distant by more than $r>0$, is it because $\mathcal Y$ is the countable union of sets of diameters not bigger than $r$? We investigate connections between the set-theoretic phenomena involved and the geometric properties of uncountable subsets of nonseparable Banach spaces of densities up to $2^\omega$ related to uncountable $(1+)$-separated sets, equilateral sets or Auerbach systems. The results include geometric dichotomies for a wide range of classes of Banach spaces, some in ZFC, some under the assumption of OCA+MA and some under a hypothesis on the descriptive complexity of the space as well as constructions (in ZFC or under CH) of Banach spaces where the geometry of the unit sphere displays anti-Ramsey properties. This complements classical theorems for separable spheres and the recent results of H\'ajek, Kania, Russo for densities above $2^\omega$ as well as offers a synthesis of possible phenomena and categorization of examples for uncountable densities up to $2^\omega$ obtained previously by the author and Guzm\'an, Hru\v{s}\'ak, Ryduchowski and Wark. It remains open if the dichotomies may consistently hold for all Banach spaces of the first uncountable density or if the strong anti-Ramsey properties of the distance on the unit sphere of a Banach space can be obtained in ZFC.