  For any set $S$ 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 this paper, we determine the minimum
size of 3-uniform bi-hypergraphs which are one-realizations of a given set $S$.
As a result, we partially solve an open problem proposed by Bujt$\acute{\rm
a}$s and Tuza in 2008.
