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arXiv · 1905.00782

Uncountable dichromatic number without short directed cycles

Attila Joó
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Abstract

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.

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Attila Joó. 2019-05-02. Uncountable dichromatic number without short directed cycles. https://arxiv.org/abs/1905.00782

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