Two results on the digraph chromatic number

It is known (Bollobás (1978) [4]; Kostochka and Mazurova (1977) [12]) that there exist graphs of maximum degree ΔΔ and of arbitrarily large girth whose chromatic number is at least cΔ/logΔcΔ/logΔ. We show an analogous result for digraphs where the chromatic number of a digraph DD is defined as the minimum integer kk so that V(D)V(D) can be partitioned into kk 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 Erdős (1962) [5], that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.