  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}$.
