Hypergraph colouring and the Lovász Local Lemma Original Research Article

The Lovász Local Lemma yields sufficient conditions for a hypergraph to be 2-colourable, that is, to have a colouring of the points blue or red so that no edge is monochromatic. The method yields a general theorem, which shows for example that, if H is a hypergraph in which each edge contains at least 9 points and each point is contained in at most 11 edges, then H is 2-colourable. Here we see that this approach can go a little further: we use the ‘lopsided’ version of the Local Lemma to give an improved version of the theorem on hypergraph 2-colouring, from which it follows for example that the numbers 9, 11 above may be replaced by 8, 12.