The smallest one-realization of a given set II

For any set SS of positive integers, a mixed hypergraph HH is a one-realization of SS if its feasible set is SS and each entry of its chromatic spectrum is either 0 or 1. In [P. Zhao, K. Diao, K. Wang, The smallest one-realization of a given set, Electron. J. Combin. 19 (2012) #P19], we determined the minimum number of vertices of one-realizations of a given set SS. In this paper, we continue this research and determine the minimum number of vertices of 3-uniform bi-hypergraphs which are one-realizations of SS. As a result, we partially solve an open problem proposed by Bujtás and Tuza in 2008.