On the Structure of Uniform One-Realizations of a Given Set

Abstract

Let \(S=\{n_1,n_2,\ldots ,n_s\}\), \(s\ge 2\), be a set of positive integers. A mixed hypergraph \({{\mathcal {H}}}=(X, {{\mathcal {C}}}, \mathcal{D})\) is a one-realization of S if for every \(i\in S\), there exists a unique coloring (up to permutations of the colors) using precisely i colors and there are no other colorings.