Non-monochromatic non-rainbow colourings of σ-hypergraphs

One of the most interesting new developments in hypergraph colourings in these last few years has been Voloshin's notion of colourings of mixed hypergraphs. In this paper we shall study a specific instance of Voloshin's idea: a non-monochromatic non-rainbow (NMNR) colouring of a hypergraph is a colouring of its vertices such that every edge has at least two vertices coloured with different colours (non-monochromatic) and no edge has all of its vertices coloured with distinct colours (non-rainbow). Perhaps the most intriguing phenomenon of such colourings is that a hypergraph can have gaps in its NMNR chromatic spectrum, that is, for some $k_1 < k_2 < k_3$, the hypergraph is NMNR colourable with $k_1$ and with $k_3$ colours but not with $k_2$ colours. Several beautiful examples have been constructed of NMNR colourings of hypergraphs exhibiting phenomena not seen in classical colourings. Many of these examples are either \emph{ad hoc} or else are based on designs. The latter are difficult to construct and they generally give uniform $r$-hypergraphs only for low values of $r$, generally $r=3$. In this paper we shall study the NMNR colourings of a type of $r$-uniform hypergraph which we call $\sigma$-hypergraphs. The attractive feature of these $\sigma$-hypergraphs is that they are easy to define, even for large $r$, and that, by suitable modifications of their parameters, they can give families of hypergraphs which are guaranteed to have NMNR spectra with gaps or NMNR spectra without gaps. These $\sigma$-hypergraphs also team up very well with the notion of colour-bounded hypergraphs recently introduced by Bujt{\'a}s and Tuza to give further control on the appearance of gaps and perhaps explain better the existence of gaps in the colouring of mixed hypergraphs.