A geometric theory of hypergraph colouring

Summary Ak-colouring of a hypergraph is an assignment of no more thank colours to the vertices of the hypergraph in such a way that the coloured hypergraph has no monochromatic edges. In this paper a polymatroid is associated with a hypergraph. It is shown that the number ofk-colourings of the hypergraph is determined by an evaluation of the characteristic polynomial of this polymatroid. Hypergraph colouring is then related to an extension of the critical problem developed by the author. It is shown that, ifq is a prime power, then a hypergraph isq^k-colourable if and only if the critical exponent overGF(q) of its associated polymatroid is less than or equal tok.