The chromatic spectrum of 3-uniform bi-hypergraphs

Let S = { n 1 , n 2 , … , n t } be a finite set of positive integers with min S ≥ 3 and t ≥ 2 . For any positive integers s 1 , s 2 , … , s t , we construct a family of 3-uniform bi-hypergraphs H with the feasible set S and r n i = s i , i = 1 , 2 , … , t , where each r n i is the n i th component of the chromatic spectrum of H . As a result, we solve one open problem for 3 -uniform bi-hypergraphs proposed by Bujtás and Tuza in 2008. Moreover, we find a family of sub-hypergraphs with the same feasible set and the same chromatic spectrum as its own. In particular, we obtain a small upper bound on the minimum number of vertices in 3-uniform bi-hypergraphs with the feasible set S .