  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 previousy 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.

Logic;Functional Analysis;General Topology