On the growth rate of dichromatic numbers of finite subdigraphs

Chris Lambie-Hanson proved recently that for every function $f:\mathbb{N}\to \mathbb{N}$ there is an ${\mathrm{\aleph }}_1$-chromatic graph $G$ of size $2^{{\mathrm{\aleph }}_1}$ 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 $\mathbb{N}$ $anuncountablydichromaticdigraph$ D $ofsize$ 2^{\aleph_0} $suchthatevery$ (n+2) $-dichromaticsubgraphof$ D $hasatleast$ 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 $2^{{\mathrm{\aleph }}_0}$” can be replaced by “$\kappa$-dichromatic” and “of size $\kappa$” respectively where $\kappa$ is universally quantified with bounds ${\mathrm{\aleph }}_0\le \kappa \le 2^{{\mathrm{\aleph }}_0}$.