Maximal trees
  We show that, consistently, there can be maximal subtrees of P (omega) and P (omega) / fin of arbitrary regular uncountable size below the size of the continuum. We also show that there are no maximal subtrees of P (omega) / fin with countable levels. Our results answer several questions of Campero, Cancino, Hrusak, and Miranda. 
Joerg Brendle
math.LO