Consistently there exist ℵ2-chromatic graphs with no ℵ1-chromatic subgraphs. The statement that every uncountably chromatic graph of size ℵ1 contains an uncountably chromaticω-connected subgraph is consistent and independent. It is consistent that there is an uncountably chromatic graph of size ℵω 1 in which every subgraph with size less than ℵω 1 is countably chromatic.
KeywordsChromatic Number Connected Subgraph Infinite Graph Homogeneous Color Common Extension
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