Almost disjoint families and the geometry of nonseparable spheres

We consider uncountable almost disjoint families of subsets of N, the Johnson-Lindenstrauss Banach spaces (X-A,parallel to parallel to(infinity)) induced by them, and their natural equivalent renormings (XA,parallel to parallel to(infinity,2)). We introduce a partial order PA and characterize some geometric properties of the spheres of (X-A,parallel to parallel to(infinity)) and of (X-A,parallel to parallel to(infinity,2)) in terms of combinatorial properties of PA. Exploiting the extreme behavior of some known and some new almost disjoint families among others we show the existence of Banach spaces where the unit spheres display surprising geometry:<br />1) There is a Banach space of density continuum whose unit sphere is the union of countably many sets of diameters strictly less than 1.<br />2) It is consistent that for every rho>0 there is a nonseparable Banach space, where for every delta>0 there is epsilon>0 such that every uncountable (1-epsilon)-separated set of elements of the unit sphere contains two elements distant by less than 1 and two elements distant at least by 2-rho-delta.<br />It should be noted that for every epsilon>0 every nonseparable Banach space has a plenty of uncountable (1-epsilon)-separated sets by the Riesz Lemma.<br />We also obtain a consistent dichotomy for the spaces of the form (X-A,parallel to parallel to(infinity,2)): The Open Coloring Axiom implies that the unit sphere of every Banach space of the form (X-A,parallel to parallel to(infinity,2)) either is the union of countably many sets of diameter strictly less than 1 or it contains an uncountable (2-epsilon)-separated set for every epsilon>0.