On the growth rate of dichromatic numbers of finite subdigraphs

Attila Joó

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arXiv · 1908.07264

On the growth rate of dichromatic numbers of finite subdigraphs

Attila Joó

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Abstract

Chris Lambie-Hanson proved recently that for every function $ f:\mathbb{N}\rightarrow \mathbb{N} $ there is an $ \aleph_1 $-chromatic graph $ G $ of size $ 2^{\aleph_1} $ such that every $ (n+3) $-chromatic subgraph of $ G $ has at least $ f(n) $ vertices. Previously, this fact was just known to be consistently true due to P. Komj\'ath and S. Shelah. We investigate the analogue of this question for directed graphs. In the first part of the paper we give a simple method to construct for an arbitrary $ f:\mathbb{N}\rightarrow \mathbb{N} $ an uncountably dichromatic digraph $ D $ of size $ 2^{\aleph_0} $ such that every $ (n+2) $-dichromatic subgraph of $ D $ has at least $ f(n) $ vertices. In the second part we show that it is consistent with arbitrary large continuum that in the previous theorem "uncountably dichromatic" and "of size $ 2^{\aleph_0} $" can be replaced by "$\kappa $-dichromatic" and "of size $ \kappa $" respectively where $ \kappa $ is universally quantified with bounds $ \aleph_0 \leq \kappa \leq 2^{\aleph_0}$.

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Attila Joó. 2019-08-20. On the growth rate of dichromatic numbers of finite subdigraphs. https://arxiv.org/abs/1908.07264

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