Consistency results on infinite graphs
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 ℵω1 in which every subgraph with size less than ℵω1 is countably chromatic.
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