Chain method for panchromatic colorings of hypergraphs

Abstract

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 $3<r\le \sqrt[3]{n/(100\ln n)}$, every $n$-uniform hypergraph $H$ with $|E(H)|\le \frac{1}{20r^2}{\left(\frac{n}{\ln n}\right)}^{\frac{r-1}{r}}{\left(\frac{r}{r-1}\right)}^{n-1}$ has a panchromatic coloring with $r$ colors.