Uncountable dichromatic number without short directed cycles

Abstract

A. Hajnal and P. 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 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.

arXiv comment3 pages, 1 figure