The chromatic spectrum of 3-uniform bi-hypergraphs

Let S={n1,n2,…,nt}S={n1,n2,…,nt} be a finite set of positive integers with minS≥3minS≥3 and t≥2t≥2. For any positive integers s1,s2,…,sts1,s2,…,st, we construct a family of 3-uniform bi-hypergraphs HH with the feasible set SS and rni=si,i=1,2,…,trni=si,i=1,2,…,t, where each rnirni is the ninith component of the chromatic spectrum of HH. As a result, we solve one open problem for 33-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 SS.