FUNDAMENTA MATHEMATICAE 234 (2016) Incomparable families and maximal trees by G. Campero-Arena (M´exico, D.F.), J. Cancino (Morelia), M. Hruˇs´ak (M´exico, D.F.) and F. E. Miranda-Perea (M´exico, D.F.) Abstract. 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. 1. Introduction. A chain in a partially ordered set (P, ≤) is a subset of P which is linearly ordered by ≤. On the other hand, the term antichain in P has two, quite different yet commonly used, meanings: in forcing terminol- ogy, an antichain is a set of elements of P any two of which are mutually in- compatible (i.e. have no common lower bound); the other refers to families of pairwise incomparable elements. We shall call the former antichains and the latter incomparable families. We shall always assume that an incomparable family does not contain the maximal element of P, which we require to exist. Similarly, there are two distinct notions of a subtree of a partially ordered set P (for their connection with forcing “growing downward”). We call a partially ordered set (T, ≤)a tree if it has a largest element 1 and for every t ∈ T the set pred T (t)= {s ∈ T : s ≥ t} is well-ordered by the reverse order of ≤, i.e. pred T (t) is linearly ordered by ≤ with every strictly increasing chain being finite. Accordingly, T ⊆ P is a subtree of a partially ordered set (P, ≤) with a maximal element 1 if 1 ∈ T and (T, ≤(T × T)) is a tree. Note that we do not require that incomparable (equivalently, incompatible) elements of T are incompatible in the partial order P ( 1 ). 2010 Mathematics Subject Classification : 03E17, 03E35, 03G05. Key words and phrases : Boolean algebra, antichain, incomparable family, tree, cardinal invariant. Received 6 May 2015; revised 13 August 2015. Published online 20 January 2016. ( 1 ) Our notation for Boolean algebras differs from that of Monk [14] in that our trees are exactly images of trees according to Monk by the map which sends each element of the Boolean algebra to its complement. DOI: 10.4064/fm125-1-2016 [73] c  Instytut Matematyczny PAN, 2016