  The historical first detection of a binary neutron star merger by the
LIGO-Virgo collaboration [B. P. Abbott et al. Phys. Rev. Lett. 119, 161101
(2017)] is providing fundamental new insights into the astrophysical site for
the $r$-process and on the nature of dense matter. A set of realistic models of
the equation of state (EOS) that yield an accurate description of the
properties of finite nuclei, support neutron stars of two solar masses, and
provide a Lorentz covariant extrapolation to dense matter are used to confront
its predictions against tidal polarizabilities extracted from the
gravitational-wave data. Given the sensitivity of the gravitational-wave signal
to the underlying EOS, limits on the tidal polarizability inferred from the
observation translate into constraints on the neutron-star radius. Based on
these constraints, models that predict a stiff symmetry energy, and thus large
stellar radii, can be ruled out. Indeed, we deduce an upper limit on the radius
of a $1.4\,M_{\odot}$ neutron star of $R_{\star}^{1.4}\!<\!13.76\,{\rm km}$.
Given the sensitivity of the neutron-skin thickness of ${}^{208}$Pb to the
symmetry energy, albeit at a lower density, we infer a corresponding upper
limit of about $R_{\rm skin}^{208}\!\lesssim\!0.25\,{\rm fm}$. However, if the
upcoming PREX-II experiment measures a significantly thicker skin, this may be
evidence of a softening of the symmetry energy at high densities---likely
indicative of a phase transition in the interior of neutron stars.
