We prove: Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a κ-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph into G. Theorem 2. In the Solovay model, every OD graph G on reals satisfies one of the following two requirements: (I) G admits an OD colouring by countable ordinals; (II) as above.
@article{bwmeta1.element.bwnjournal-article-fmv154i2p183bwm,
author = {Vladimir Kanovei},
title = {Two dichotomy theorems on colourability of non-analytic graphs},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {183-201},
zbl = {0883.03035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv154i2p183bwm}
}
Kanovei, Vladimir. Two dichotomy theorems on colourability of non-analytic graphs. Fundamenta Mathematicae, Tome 154 (1997) pp. 183-201. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv154i2p183bwm/
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