The smallest 3-uniform bi-hypergraphs which are one-realization of a given set

TL;DR

This paper determines the minimal size of 3-uniform bi-hypergraphs that exactly realize a given set of positive integers as their feasible set, addressing an open problem in hypergraph theory.

Contribution

It identifies the smallest 3-uniform bi-hypergraphs that serve as one-realizations for any specified set, advancing understanding of hypergraph realizations.

Findings

Established the minimum size for 3-uniform bi-hypergraphs as one-realizations.

Partially solved an open problem from 2008.

Provided a characterization for minimal hypergraph constructions.

Abstract

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ 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 this paper, we determine the minimum size of 3-uniform bi-hypergraphs which are one-realizations of a given set $S$. As a result, we partially solve an open problem proposed by Bujt$\acute{\rm a}$s and Tuza in 2008.