Chris Lambie-Hanson proved recently that for every function f : N -> 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. Komjath 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 : N -> 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 arbitrarily 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) <= kappa <= 2(aleph 0). (C) 2019 Elsevier B.V. All rights reserved.