  In this paper we prove parts of a conjecture of Herzog giving lower bounds on
the rank of the free modules appearing in the linear strand of a graded $k$-th
syzygy module over the polynomial ring. If in addition the module is
$\mathbb{Z}^n$-graded we show that the conjecture holds in full generality.
Furthermore, we give lower and upper bounds for the graded Betti numbers of
graded ideals with a linear resolution and a fixed number of generators.
