Uncountable dichromatic number without short directed cycles

Hajnal and Erdős 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 Soukup that, in contrast of the undirected case, it is consistent that for any n < ω there exists an uncountably dichromatic digraph without directed cycles shorter than n . He asked if it is provable already in ZFC (i.e., Zermelo ‐Fraenkel set theory with the Axiom of choice). We answer his question positively by constructing for every infinite cardinal κ and n < ω a digraph of size 2 κ with dichromatic number at least κ + without directed cycles of length less than n .