Ramsey properties of the distance on nonseparable spheres

Given a nonseparable metric space (M,d) bounded by 2 we consider
(A) dichotomies D_r for r in (0,2): either there is uncountable subset N of M such that d(x, y) > r for all distinct x,y  in N or else M is the union of countably many sets each of diameter not bigger than r;
(B) the metric failure of the Ramsey property at the first uncontable cardinal: there is e > 0 such that for every uncountable subset N of M there are  distinct  x,y, u,v in N such that |d(x, y) - d(u, v)| > e.

I turns out that the general situation reduces to subsets of the unit spheres of Banach spaces with the norm distance and that set-theoretic consideration translate to well known geometric problems.  Results include ZFC theorems and consequences of OCA, MA or CH as well as results depending on the descriptive complexity of the the geometric structures.
It remains open if it is consistent that all unit spheres of density omega_1 satisfy D_r for all but one r in (0,2) and if one can construct in ZFC a nonseparable unit sphere satisfying (B) for every separated uncountable subset.

The next seminar meeting will be held on June 14.