It is consistent that but .
@article{bwmeta1.element.bwnjournal-article-fmv155i1p95bwm, author = {P\'eter Komj\'ath}, title = {Distinguishing two partition properties of $\omega$1}, journal = {Fundamenta Mathematicae}, volume = {158}, year = {1998}, pages = {95-99}, zbl = {0894.03023}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv155i1p95bwm} }
Komjáth, Péter. Distinguishing two partition properties of ω1. Fundamenta Mathematicae, Tome 158 (1998) pp. 95-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv155i1p95bwm/
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