  Consider a directed graph (digraph) in which vertices are assigned color
sets, and two vertices are connected if and only if they share at least one
color and the tail vertex has a strictly smaller color set than the head. We
seek to determine the smallest possible size of the union of the color sets
that allows for such a digraph representation. To address this problem, we
introduce the new notion of a directed intersection representation of a
digraph, and show that it is well-defined for all directed acyclic graphs
(DAGs). We then proceed to introduce the directed intersection number (DIN),
the smallest number of colors needed to represent a DAG. Our main results are
upper bounds on the DIN of DAGs based on what we call the longest terminal path
decomposition of the vertex set, and constructive lower bounds.
