  The electromagnetic (EM) signal of a binary neutron star (BNS) merger depends
sensitively on the total binary mass, $M_{\rm tot}$, relative to various
threshold masses set by the neutron star (NS) equation-of-state (EOS),
parameterized through the NS maximum mass, $M_{\rm TOV}$, and characteristic
radius, $R_{1.6}$. EM observations of a BNS merger detected through its
gravitational wave (GW) emission, which are of sufficient quality to ascertain
the identity of the merger remnant, can therefore constrain the values of
$M_{\rm TOV}$ and $R_{1.6}$, given the tight connection between $M_{\rm tot}$
and the well-measured chirp mass. We elucidate the present and future landscape
of EOS constraints from BNS mergers, introducing the "Multi-Messenger Matrix",
a mapping between GW and EM measurables that defines the ranges of event chirp
masses which provide the most leverage on constraining the EOS. By simulating a
population of BNS mergers drawn from the Galactic double NS mass distribution
we show that $\sim 10$ joint detections can constrain $M_{\rm TOV}$ and
$R_{1.6}$ to several percent level where systematic uncertainties may become
significant. Current EOS constraints imply that most mergers will produce
supramassive or hypermassive remnants, a smaller minority (possibly zero) will
undergo prompt-collapse, while at most only a few percent of events will form
indefinitely stable NSs. In support of the envisioned program, we advocate in
favor of LIGO/Virgo releasing chirp mass estimates as early as possible to the
scientific community, enabling observational resources to be allocated in the
most efficient way to maximize the scientific gain from multi-messenger
discoveries.
