The graph of monomial ideals

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring K[x1,...,xn]. The definition involves Gröbner bases or the action of the algebraic torus (K*)n. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on square-free monomial ideals. In the latter case, the edges correspond to generalized bistellar flips.