  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.
