Almost disjoint families and the geometry of spheres

We present an example of a nonseparable Banach space where every uncountable subset of the unit sphere contains two vectors which are distant by less than one. This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which has produced many deep partial positive results. In fact, the space has many more peculiar new properties related to equilateral sets, Auerbach systems, packing the unit ball or others. The space is constructed with the help of an uncountable almost disjoint family of subsets of N with some special properties. The arguments are relatively nontechnical. The preprint is available at: https://arxiv.org/pdf/2104.05335.pdf