It is consistent that there exists a graph X of cardinality such that every graph has an edge coloring with colors in which the induced copies of X (if there are any) are totally multicolored (get all possible colors).
@article{bwmeta1.element.bwnjournal-article-fmv154i2p203bwm,
author = {P\'eter Komj\'ath},
title = {A strongly non-Ramsey uncountable graph},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {203-205},
zbl = {0883.03037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv154i2p203bwm}
}
Komjáth, Péter. A strongly non-Ramsey uncountable graph. Fundamenta Mathematicae, Tome 154 (1997) pp. 203-205. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv154i2p203bwm/
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