A note on the minimum number of edges in hypergraphs with property O

Abstract

An oriented $k$-graph 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 Rödl investigated the minimum number $f\left(k\right)$ of edges in a $k$-uniform hypergraph with Property O. They proved that $k!\le f\left(k\right)\le (k^{2}lnk)k!$, where the upper bound holds for sufficiently large $k$. In this short note we improve their upper bound by a factor of $klnk$ showing that $f\left(k\right)\le \left(\lfloor \frac{k}{2}\rfloor +1\right)k!-\lfloor \frac{k}{2}\rfloor (k-1)!$ for every $k\ge 3$. We also show that their lower bound is not tight. Furthermore, Duffus, Kay and Rödl also studied the minimum possible number $n\left(k\right)$ of vertices in an oriented $k$-graph with Property O. For $k=3$ they showed that $n\left(3\right)\in \{6,7,8,9\}$, and asked for the precise value of $n\left(3\right)$. Here we show that $n\left(3\right)=6$.