  Inspired by earlier results about proper and polychromatic coloring of
hypergraphs, we investigate such colorings of directed hypergraphs, that is,
hypergraphs in which the vertices of each hyperedge is partitioned into two
parts, a tail and a head. We present a conjecture of D. P\'alv\"olgyi and the
author, which states that directed hypergraphs with a certain restriction on
their pairwise intersections can be colored with two colors. Besides other
contributions, our main result is a proof of this conjecture for $3$-uniform
directed hypergraphs. This result can be phrased equivalently such that if a
$3$-uniform directed hypergraph avoids a certain directed hypergraph with two
hyperedges, then it admits a proper $2$-coloring. Previously, only extremal
problems regarding the maximum number of edges of directed hypergraphs that
avoid a certain hyperedge were studied.
