  The binding of polarons, or its absence, is an old and subtle topic. Here we
prove two things rigorously. First, the transition from many-body collapse to
the existence of a thermodynamic limit for N polarons occurs precisely at
U=2\alpha, where U is the electronic Coulomb repulsion and \alpha is the
polaron coupling constant. Second, if U is large enough, there is no
multi-polaron binding of any kind. Considering the known fact that there is
binding for some U>2\alpha, these conclusions are not obvious and their proof
has been an open problem for some time.
