Density Conditions for Panchromatic Colourings of Hypergraphs

Let H=(V,E) be a hypergraph. A panchromatic t-colouring of H is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and H is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of H is h(H) =min {∣∣⋃F∣∣/|F| :∅ =F⊆E}. Among other results, it is proved here that if every edge has at least t vertices and ∣∣⋃F∣∣ (t−1)|F|− t+3 whenever ∅ =F⊆E , then H is panchromatically t-choosable, and this condition is sharp; the minimum ct such that every t-uniform hypergraph with h(H)>ct is panchromatically t-choosable satisfies t−2+3/(t+1) ct t−2+4/(t+2); and except possibly when t= 3 or 5, a t-uniform hypergraph is panchromatically t-colourable if ∣∣⋃F∣∣ ((t − 2t+2)|F|+ t− 1)/t whenever ∅ =F ⊆E , and this condition is sharp. This last result dualizes to a sharp sufficient condition for the chromatic index of a hypergraph to equal its maximum degree.