The smallest one-realization of a given set II

For any set S of positive integers, a mixed hypergraph 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, 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 S . In this paper, we continue this research and determine the minimum number of vertices of 3-uniform bi-hypergraphs which are one-realizations of S . As a result, we partially solve an open problem proposed by Bujtلs and Tuza in 2008.