  We deal with an extremal problem concerning panchromatic colorings of
hypergraphs. A vertex $r$-coloring of a hypergraph $H$ is \emph{panchromatic}
if every edge meets every color. We prove that for every
$3<r\leq\sqrt[3]{n/(100\ln n)}$, every $n$-uniform hypergraph $H$ with
$|E(H)|\leq \frac{1}{20r^2}\left(\frac{n}{\ln n}\right)^{\frac
{r-1}{r}}\left(\frac{r}{r-1}\right)^{n-1}$ has a panchromatic coloring with $r$
colors.
