Almost disjoint families and the geometry of nonseparable spheres

Banach Spaces in Which Large Subsets of Spheres Concentrate

“…This is necessary as there are ZFC examples which satisfy the second part of the dichotomy and not the first (Propositions 46 and 70). On the other hand the dichotomy of Theorem 4 generalizes the dichotomies for Johnson-Lindenstrauss spaces of and for some of their renormings in [23]. Parts of the content of Theorems 3 (2), (3) include and generalize the argument of Erdös and Preiss ( [10]) that the unit ball of ℓ 2 (2 ω ) is the countable union of sets of diameters less than √ 2 + ε for every ε > 0 and a result of Preiss and Rödl ([40] that it is not the countable union of sets of diameters not bigger than √ 2.…”

“…Both of the questions were answered recently in the negative in ZFC in [33] and in [34], [33] respectively and stronger negative solutions were obtained consistently in [35] and [23] . The spaces are of density 2 ω .…”

“…The spaces are of density 2 ω . All these examples, including the above ZFC examples enjoy rather new metric phenomena in nonseparable Banach spaces, e.g., some of them admit no uncountable, or even no infinite equilateral sets ( [34]), the unit spheres of some of them are the unions of countably many sets of diameters strictly less than 1 ( [23]) or some of them admit no uncountable Auerbach systems ( [33]; for definition of Auerbach systems see Section 2.1; first such example was obtained under CH in [25]). At the same time other results showed that the above-mentioned stronger consistent irregularity properties of Banach spaces of particular types are consistently impossible: all Banach spaces of the form C(K) admit uncountable equilateral sets under MA+¬CH (Theorem of 5.1 [31]), there are geometric dichotomies for Johnson-Lindenstrauss spaces and for some of their renormings under OCA (Theorem 2 of [23]), the spheres of Banach spaces of Shelah induced by anti-Ramsey colorings still admit uncountable (1+)-separated and equilateral sets under MA+¬CH (Theorem 3 (3) of [35]).…”

“…is the union of countably many sets of diameter not bigger than r. This is, for example, the case for the spaces of [33] and [23] with no uncountable (1+)-separated sets.…”

“…It should be also clear that being hyperlateral corresponds to anti-Ramsey properties of Sierpiński's or Todorcevic's colorings (Theorems 29, 31) as any uncountable separated sets hits many colors-distances, where many means with fixed nonzero oscillation (fixed for a given separated set where subsets are considered). Our results on hyperlateral Banach spaces consists of proving (in some cases, just noting) that some spaces considered in the papers [23,31,35] are hyperlateral and exploring their additional properties in this context. Theorem 8.…”