# A Note on the Minimum Number of Edges in Hypergraphs with Property O

## TL;DR

This paper refines bounds on the minimum number of edges in hypergraphs with Property O, providing a tighter upper bound and determining the exact minimum number of vertices for the case k=3.

## Contribution

It improves the upper bound on the minimum edges in hypergraphs with Property O and determines the exact minimum vertices for k=3.

## Key findings

- Improved upper bound for f(k) by a factor of k ln k.
- Showed that the lower bound for f(k) is not tight.
- Established that n(3)=6 for the minimum vertices in 3-uniform hypergraphs with Property O.

## Abstract

An oriented $k$-uniform hypergraph is said to have Property O if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. Recently Duffus, Kay and R\"{o}dl investigated the minimum number $f(k)$ of edges in a $k$-uniform hypergaph with Property O. They proved that $k! \leq f(k) \leq (k^2 \ln k) k!$, where the upper bound holds for $k$ sufficiently large. In this short note we improve their upper bound by a factor of $k \ln k$, showing that $f(k) \le \left(\lfloor \frac{k}{2} \rfloor +1 \right) k! - \lfloor \frac{k}{2} \rfloor (k-1)!$ for every $k\geq 3$. We also show that their lower bound is not tight. Furthermore, Duffus, Kay and R\"{o}dl also studied the minimum number $n(k)$ of vertices in a $k$-uniform hypergaph with Property O. For $k=3$ they showed $n(3) \in \{6,7,8,9\}$, and asked for the precise value of $n(3)$. Here we show $n(3)=6$.