  Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a
field $K$ with maximal ideal $\frak{m}=(x_1,...,x_n)$, and let $I$ be a graded
ideal of $S$. In this paper, we define the saturation number $\sat(I)$ of $I$
to be the smallest non-negative integer $k$ such that $I:\mm^{k+1}= I:\mm^k$.
We show that $f(k)$ is linearly bounded, and that $f(k)$ is a quasi-linear
function for $k\gg 0$, if $I$ is a monomial ideal. Furthermore, we show that
$\sat(I^k)=k$ if $I$ is a principal Borel ideal and prove that $\sat(I_{d,n}^k)
=\max\{l\:\; (kd-l)/(k-l) \leq n\},$ where $I_{d,n}$ is the squarefree Veronese
ideal generated in degree $d$. \end{abstract}
