  We analyze the vanishing and non-vanishing behavior of the graded Betti
numbers for Segre embeddings of products of projective spaces. We give lower
bounds for when each of the rows of the Betti table becomes non-zero, and prove
that our bounds are tight for Segre embeddings of products of P^1. This
generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for
Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for
computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing
of Betti numbers, and the theory of algebras with straightening laws.
