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 panchromatic if every edge meets every color. We prove that for every 2 ≤ r < n 100 ln n 3 , every n -uniform hypergraph H with | E ( H ) | ≤ c r 2 n ln n r − 1 r r r − 1 n − 1 has a panchromatic coloring with r colors, where c > 0 is an absolute constant.