  Let a 2 to 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 nth 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.
There are actually two versions of this problem: the standard version where
every triple of vertices is allowed to have up to all three possible directed
edges and the oriented version where each triple can have at most one directed
edge. In this paper, we determine the standard extremal numbers and the
oriented extremal numbers for three different directed hypergraphs. Each has
exactly two edges, and of the seven (nontrivial) 2 to 1 graphs with exactly two
edges, these are the only three with extremal numbers that are cubic in n. The
standard and oriented extremal numbers for the other four directed hypergraphs
with two edges are determined in a companion paper.
