  We study Hilbert generated versions of nonseparable Banach spaces $\mathcal
X$ considered by Shelah, Stepr\=ans and Wark where the behavior of the norm on
nonseparable subsets is so irregular that it does not allow any linear bounded
operator on $\mathcal X$ other than a diagonal operator (or a scalar multiple
of the identity) plus a separable range operator. We address the questions if
these spaces admit uncountable equilateral sets and if their unit spheres admit
uncountable $(1+)$-separated or $(1+\varepsilon)$-separated sets. We resolve
some of the above questions for two types of these spaces by showing both
absolute and undecidability results.
  The corollaries are that the continuum hypothesis (in fact: the existence of
a nonmeager set of reals of the first uncountable cardinality) implies the
existence of an equivalent renorming of the nonsepareble Hilbert space
$\ell_2(\omega_1)$ which does not admit any uncountable equilateral set and it
implies the existence of a nonseparable Hilbert generated Banach space
containing an isomorphic copy of $\ell_2$ in each nonseparable subspace, whose
unit sphere does not admit an uncountable equilateral set and does not admit an
uncountable $(1+\varepsilon)$-separated set for any $\varepsilon>0$. This could
be compared with a recent result by H\'ajek, Kania and Russo saying that all
nonseparable reflexive Banach spaces admit uncountable
$(1+\varepsilon)$-separated sets in their unit spheres.
