(2,2)(2,2)-colourings and clique-free σσ-hypergraphs

We consider vertex colourings of rr-uniform hypergraphs HH in the classical sense, that is such that no edge has all its vertices given the same colour, and (2,2)(2,2)-colourings of HH in which the vertices in any edge are given exactly two colours. This is a special case of constrained colourings introduced by Bujtas and Tuza which, in turn, is a generalization of Voloshin’s colourings of mixed hypergraphs. We study, χ(H)χ(H), the classical chromatic number, and the (2,2)(2,2)-spectrum of HH, that is, the set of integers kk for which HH has a (2,2)(2,2)-colouring using exactly kk colours.We present extensions of hypergraphs which preserve both the chromatic number and the (2,2)(2,2)-spectrum and which, however often repeated, do not increase the clique number of HH by more than a fixed number. In particular, we present sparse (2,2)(2,2)-colourable clique-free σσ-hypergraphs having arbitrarily large chromatic number—these rr-uniform hypergraphs were studied by the authors in earlier papers. We use these ideas to extend some known 33-uniform hypergraphs which exhibit a (2,2)(2,2)-spectrum with remarkable gaps. We believe that this work is the first to present an extension of hypergraphs which preserves both χ(H)χ(H) and the (2,2)(2,2)-spectrum of HH simultaneously.