  Let $S$ be a finite set of positive integers. A mixed hypergraph ${\cal H}$
is a one-realization of $S$ if its feasible set is $S$ and each entry of its
chromatic spectrum is either 0 or 1. In [P. Zhao, K. Diao, Y. Chang and K.
Wang, The smallest one-realization of a given set
\uppercase\expandafter{\romannumeral2}, Discrete Math. 312 (2012) 2946--2951],
we determined the minimum number of vertices of a 3-uniform bi-hypergraph which
is a one-realization of $S$. In this paper, we generalize this result to
$r$-uniform bi-hypergraphs.
