  It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there
exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose
chromatic number is at least $c \Delta / \log \Delta$. We show an analogous
result for digraphs where the chromatic number of a digraph $D$ is defined as
the minimum integer $k$ so that $V(D)$ can be partitioned into $k$ acyclic
sets, and the girth is the length of the shortest cycle in the corresponding
undirected graph. It is also shown, in the same vein as an old result of Erdos
(1962), that there are digraphs with arbitrarily large chromatic number where
every large subset of vertices is 2-colorable.

Combinatorics