We will briefly discuss some questions concerning the saturation
properties of models of strong forcing axioms (MM, PFA, etc).
Caicedo and Velickovic conjectured the following:
Assume MM and let W be an inner model with the same cardinals of the
universe V. Then all sequences of ordinals of length aleph_1 are already
elements of the inner model W.
We shall discuss the motivations and the ground for the above conjecture
as well as some partial results relating the conjecture to certain Ramsey
properties and to Shelah's pcf theory.