  For a Banach space $X$ its subset $Y\subseteq X$ is called overcomplete if
$|Y|=dens(X)$ and $Z$ is linearly dense in $X$ for every $Z\subseteq Y$ with
$|Z|=|Y|$. In the context of nonseparable Banach spaces this notion was
introduced recently by T. Russo and J. Somaglia but overcomplete sets have been
considered in separable Banach spaces since the 1950ties.
  We prove some absolute and consistency results concerning the existence and
the nonexistence of overcomplete sets in some classical nonseparable Banach
spaces. For example: $c_0(\omega_1)$, $C([0,\omega_1])$,
$L_1(\{0,1\}^{\omega_1})$, $\ell_p(\omega_1)$, $L_p(\{0,1\}^{\omega_1})$ for
$p\in (1, \infty)$ or in general WLD Banach spaces of density $\omega_1$ admit
overcomplete sets (in ZFC). The spaces $\ell_\infty$, $\ell_\infty/c_0$, spaces
of the form $C(K)$ for $K$ extremally disconnected, superspaces of
$\ell_1(\omega_1)$ of density $\omega_1$ do not admit overcomplete sets (in
ZFC). Whether the Johnson-Lindenstrauss space generatedin $\ell_\infty$ by
$c_0$ and the characteristic functions of elements of an almost disjoint family
of subsets of $\mathbb N$ of cardinality $\omega_1$ admits an overcomplete set
is undecidable. The same refers to all nonseparable Banach spaces with the dual
balls of density $\omega_1$ which are separable in the weak$^*$ topology. The
results proved refer to wider classes of Banach spaces but several natural open
questions remain open.
