Metric dichotomies and antiramsey geometries of nonseparable spheres
Given an uncountable subset \(Y\) of the unit sphere of a nonseparable Banach space, is there an uncountable \(Z\) included in \(Y\) such that the distances between any two distinct points of \(Z\) are more or less the same? If an uncountable subset Y of such a sphere does not admit an uncountable \(Z\) included in \(Y\), where any two points are distant by more than \(r > 0\), is it because \(Y\) is the countable union of sets of diameters not bigger than \(r\)?Clearly, these types of questions can be rephrased in the combinatorial language of partitions of pairs of points of a Banach space \(X\) induced by the distance function. We investigate connections between the set-theoretic phenomena involved (both descriptive and combinatorial) and the geometric properties of uncountable subsets of the unit spheres of nonseparable Banach spaces of densities up to continuum. This investigation is related to classical topics concerning uncountable \((1+)\)-separated sets, equilateral sets or Auerbach systems but also yields rather new opening on metric phenomena in nonseparable spheres.
Extensive references to the literature can be found in the paper P. Koszmider: On Ramsey-type properties of the distance in nonseparable spheres, accepted to Trans. Amer. Math. Soc. with the preprint available at https://arxiv.org/pdf/2308.07668.