2019/08/20 by Joó, Attila
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1908.07264
Chris Lambie-Hanson proved recently that for every function f:ℕ→ ℕ there is an ℵ1 -chromatic graph G of size 2ℵ1 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. Komjáth 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 2ℵ0 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 2ℵ0 " can be replaced by "κ-dichromatic" and "of size κ" respectively where κ is universally quantified with bounds ℵ0 ≤ κ≤ 2ℵ0.