Claims

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1984_03_01_baumgartner: Baumgartner (J. Symbolic Logic, 1984) proves it consistent with ZFC and GCH that a graph on aleph two vertices has chromatic number aleph two while all its subgraphs on at most aleph one vertices are countably chromatic.

1988_02_01_foreman_laver: Foreman and Laver (Adv. Math., 1988) build from a huge cardinal a model of ZFC and GCH in which every graph of size and chromatic number aleph two has a subgraph of size and chromatic number aleph one.

2014_08_05_rinot: Rinot (Combinatorica, 2015) proves from square at lambda and 2^lambda = lambda^+ a graph on lambda^+ vertices of chromatic number aleph one whose smaller subgraphs are countably chromatic; at lambda = aleph omega, in L.

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