Extremal Numbers for Directed Hypergraphs with Two Edges

Abstract

Let a $2 \rightarrow 1$ directed hypergraph be a 3-uniform hypergraph where every edge has two tail vertices and one head vertex. For any such directed hypergraph $F$, let the $n$th extremal number of $F$ be the maximum number of edges that any directed hypergraph on $n$ vertices can have without containing a copy of $F$. In 2007, Langlois, Mubayi, Sloan, and Turán determined the exact extremal number for a particular directed hypergraph and found the extremal number up to asymptotic equivalence for a second directed hypergraph. Each of these forbidden graphs had exactly two edges. In this paper, we determine the exact extremal numbers for every $2 \rightarrow 1$ directed hypergraph that has exactly two edges.