  This study addresses the collapse behavior of neutron star (NS) mergers
expressed through the binary threshold mass M_thr for prompt black hole (BH)
formation, which we determine by relativistic hydrodynamical simulations for 40
equation of state (EoS) models. M_thr can be well described by various fit
formulae involving stellar parameters of nonrotating NSs. Using these relations
we compute which constraints on NS radii and the tidal deformability are set by
current and future merger detections revealing information about the merger
product. We systematically investigate the impact of the binary mass ratio
q=M_1/M_2 and assemble different fits, which make different assumptions about
a-priori knowlegde. We find fit formulae for M_thr including an explicit q
dependence, which are valid in a broad range of 0.7<=q<=1 and which are nearly
as tight as relations for fixed mass ratios. For most EoS models except extreme
cases M_thr of asymmetric mergers is equal or smaller than the one of
equal-mass binaries. The impact of the binary mass asymmetry on M_thr becomes
stronger with more extreme mass ratios, while M_thr is approximately constant
for small deviations from q=1. We describe that a phase transition to
deconfined quark matter can leave a characteristic imprint on the collapse
behavior. The presence of quark matter can reduce the stability of the remnant
and thus M_thr relative to a purely hadronic reference model. Comparing the
threshold mass and the tidal deformability Lambda_thr of a system with M_thr
can yield peculiar combinations of those two quantities, where M_thr is
particularly small in relation to Lambda_thr. Hence, a combined measurement of
both quantities can indicate the onset of quark deconfinement. We point out new
univariate relations between M_thr and stellar properties of high-mass NSs,
which can be employed for direct EoS constraints or consistency checks.
(abridged)
