The chromatic spectrum of 3-uniform bi-hypergraphs

Let $S=\{n_1,n_2,...,n_t\}$ be a finite set of positive integers with $\min(S)\geq 3$ and $t\geq 2$. For any positive integers $s_1,s_2,...,s_t$, we construct a family of 3-uniform bi-hypergraphs ${\cal H}$ 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 ${\cal H}$. 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 it's own. In particular, we obtain a small upper bound on the minimum number of vertices in 3-uniform bi-hypergraphs with any given feasible set.

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