We answer several questions of D. Monk by showing that every maximal family of pairwise incomparable elements of 𝒫(ω)/fin has size continuum, while it is consistent with the negation of the Continuum Hypothesis that there are maximal subtrees of both 𝒫(ω) and 𝒫(ω)/fin of size ω₁.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm125-1-2016,
author = {G. Campero-Arena and J. Cancino and M. Hru\v s\'ak and F. E. Miranda-Perea},
title = {Incomparable families and maximal trees},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {73-89},
zbl = {06602782},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm125-1-2016}
}
G. Campero-Arena; J. Cancino; M. Hrušák; F. E. Miranda-Perea. Incomparable families and maximal trees. Fundamenta Mathematicae, Tome 233 (2016) pp. 73-89. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm125-1-2016/