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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.00782
 Bibcode:
 2019arXiv190500782J
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 3 pages, 1 figure