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$(2,2)$-colourings and clique-free $σ$-hypergraphs

We consider vertex colourings of $r$-uniform hypergraphs $H$ in the classical sense, that is such that no edge has all its vertices given the same colour, and $(2,2)$-colourings of $H$ 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 generalisation of Voloshin's colourings of mixed hypergraphs. We study, $χ(H)$, the classical chromatic number, and the $(2,2)$-spectrum of $H$, that is, the set of integers $k$ for which $H$ has a $(2,2)$-colouring using exactly $k$ colours. We present extensions of hypergraphs which preserve both the chromatic number and the $(2,2)$-spectrum and which, however often repeated, do not increase the clique number of $H$ by more than a fixed number. In particular, we present sparse $(2,2)$-colourable clique-free $σ$-hypergraphs having arbitrarily large chromatic number - these $r$-uniform hypergraphs were studied by the authors in earlier papers. We use these ideas to extend some known $3$-uniform hypergraphs which exhibit a $(2,2)$-spectrum with remarkable gaps. We believe that this work is the first to present an extension of hypergraphs which preserves both $χ(H)$ and the $(2,2)$-spectrum of $H$ simultaneously.