Incomparable families and maximal trees
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.