Abstract Observations of the centres of galaxies continue to evolve, and it is useful to take a fresh look at the constraints that exist on alternatives to supermassive black holes at their centres. We discuss constraints complementary to those of Maoz and demonstrate that an extremely wide range of other possibilities can be excluded. In particular, we present the new argument that for the velocity dispersions inferred for many galactic nuclei, even binaries made of point masses cannot stave off core collapse because hard binaries are so tight that they merge via emission of gravitational radiation before they can engage in three-body or four-body interactions. We also show that under these conditions core collapse leads inevitably to runaway growth of a central black hole with a significant fraction of the initial mass, regardless of the masses of the individual stars. For clusters of non-interacting low-mass objects (from low-mass stars to elementary particles), the relaxation of stars and compact objects that pass inside the dark region will be accelerated by interactions with the dark mass. If the dark region is instead a self-supported object, such as a fermion ball, then if stellar-mass black holes exist they will collide with the object, settle, and consume it. The net result is that the keyhole through which alternatives to supermassive black holes must pass is substantially smaller and more contrived than it was even a few years ago.