On the growth rate of dichromatic numbers of finite subdigraphs

TL;DR

This paper extends results on the growth of chromatic numbers in graphs to directed graphs, constructing large uncountably dichromatic digraphs with prescribed subgraph properties and exploring their consistency across different sizes.

Contribution

It introduces a simple construction method for uncountably dichromatic digraphs with specified subgraph size properties and analyzes their existence across various cardinalities.

Findings

Constructed uncountably dichromatic digraphs with prescribed subgraph sizes.

Proved the consistency of these constructions for various infinite cardinalities.

Extended the known results from undirected graphs to directed graphs.

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}$.