Abstract: 
We present two ${\sym P}_{\max}$ varations
which create maximal models relative to certain counterexamples
to Martin's Axiom, in hope of separating certain classical
statements which fall between MA and Suslin's Hypothesis. One of
these models is taken from $[19]$, in which we maximize relative
to the existence of a certain type of Suslin tree, and then
force with that tree. In the resulting model, all Aronszajn
trees are special and Knaster's forcing axiom ${\cal
K}_{3}$ fails. Of particular interest is the still open question
whether ${\cal K}_{2}$ holds in this model.

1. Department of Mathematics
   University of Toronto
   Toronto M5S 1A1, Canada
   
2. C.N.R.S. (7056)
   Université Paris VII
   75251 Paris Cedex 05, France