  A. Hajnal and P. Erd\H{o}s proved that a graph with uncountable chromatic
number cannot avoid short cycles, it must contain for example $ C_4 $ (among
other obligatory subgraphs). It was shown recently by D. T. Soukup that, in
contrast of the undirected case, it is consistent that for any $ n<\omega $
there exists an uncountably dichromatic digraph without directed cycles shorter
than $ n $. He asked if it is provable already in ZFC. We answer his question
positively by constructing for every infinite cardinal $ \kappa $ and $
n<\omega $ a digraph of size $ 2^{\kappa} $ with dichromatic number at least $
\kappa^{+} $ which does not contain directed cycles of length less than $ n $
as a subdigraph.
