The existence of panchromatic colourings for uniform hypergraphs

The well-known extremal problem concerning panchromatic colouring of hypergraphs is investigated. An $r$-colouring of the set of vertices of a hypergraph is said to be panchromatic if each link of the hypergraph is incident to vertices of all colours. The quantity $p(n,r)$ defined as the minimum number of links in an $n$-uniform hypergraph which admits no panchromatic $r$-colourings is studied. New lower and upper bounds for $p(n,r)$ are obtained, which improve earlier results for many of the relations between the parameters $n$ and $r$. In addition, a sufficient condition for the existence of a panchromatic $r$-colouring of an $n$-uniform hypergraph is obtained.
Bibliography: 18 items.