Decomposing 1-Sperner hypergraphs

TL;DR

This paper introduces and characterizes 1-Sperner hypergraphs, a new class with unique properties, including linear independence of hyperedge vectors, and shows they are both threshold and equilizable, with implications for combinatorics and graph theory.

Contribution

The paper defines 1-Sperner hypergraphs, provides a decomposition theorem, and establishes their key properties and bounds, advancing understanding of Sperner hypergraph classes.

Findings

1-Sperner hypergraphs are characterized by a specific set difference property.

They have linearly independent characteristic vectors over the reals.

They are both threshold and equilizable hypergraphs.

Abstract

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper.