Chromatic numbers of graphs — large gaps
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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+.
Mathematics Subject Classification (2010)03E35 05C15 05C63
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