Chromatic numbers of graphs — large gaps

We say that a graph G is (ℵ0,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ0 for any subgraph G′ of G of size < |G|.The main result of this paper reads as follows. If □λ+CHλ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ0,κ)-chromatic graph of size λ+.We also study (ℵ0,λ+)-chromatic graphs of size λ+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ0,λ+)-chromatic graph of size λ+.