  We establish a multivariate local limit theorem for the order and size as
well as several other parameters of the k-core of the Erdos-Renyi graph. The
proof is based on a novel approach to the k-core problem that replaces the
meticulous analysis of the peeling process by a generative model of graphs with
a core of a given order and size. The generative model, which is inspired by
the Warning Propagation message passing algorithm, facilitates the direct study
of properties of the core and its connections with the mantle and should
therefore be of interest in its own right.
