The Minimum Number of Edges in Uniform Hypergraphs with Property O

TL;DR

This paper investigates the minimum number of edges needed in a uniform hypergraph to ensure it has Property O, which guarantees an edge oriented consistently with any linear vertex order.

Contribution

It establishes bounds on the minimum edges required for hypergraphs to possess Property O, advancing understanding of hypergraph orientation properties.

Findings

Derived bounds on the minimum number of edges for Property O

Identified conditions under which hypergraphs exhibit Property O

Provided theoretical insights into hypergraph orientation complexity

Abstract

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.