Selective hypergraph colourings

We look at colourings of rr-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge EE in an rr-uniform hypergraph HH whose vertices are coloured is the partition of rr induced by the colour classes of the vertices in EE. Let QQ be a set of partitions of rr. A QQ-colouring of HH is a colouring of its vertices such that only patterns appearing in QQ are allowed. We first show that many known hypergraph colouring problems, including Ramsey theory, can be stated in the language of QQ-colourings. Then, using as our main tools the notions of QQ-colourings and ΣΣ-hypergraphs, we define and prove a result on tight colourings, which is a strengthening of the notion of unique colourability. ΣΣ-hypergraphs are a natural generalisation of σσ-hypergraphs introduced by the first two authors in an earlier paper. We also show that there exist ΣΣ-hypergraphs with arbitrarily large QQ-chromatic number and chromatic number but with bounded clique number. Dvořák et al. have characterised those QQ which can lead to a hypergraph with a gap in its QQ-spectrum. We give a short direct proof of the necessity of their condition on QQ. We also prove a partial converse for the special case of ΣΣ-hypergraphs. Finally, we show that, for at least one family QQ which is known to yield hypergraphs with gaps, there exist no ΣΣ-hypergraphs with gaps in their QQ-spectrum.