  These notes are extracted from the lectures on forcing axioms and
applications held by professor Matteo Viale at the University of Turin in the
academic year 2011-2012. Our purpose is to give a brief account on forcing
axioms with a special focus on some consequences of them (SRP, OCA, PID). These
principles were first isolated by Todor\v cevi\'c and interpolate most
consequences of MM and PFA, thus providing a useful insight on the
combinatorial structure of the theory of forcing axioms.
  In the first part of this notes we will give a brief account on forcing
axioms, introducing some equivalent definition by means of generalized
stationarity and presenting the consequences of them in terms of generic
absoluteness.
  In the second part we will state the strong reflection principle (SRP), prove
it under MM and examine its main consequences. This axiom is defined in terms
of reflection properties of generalized stationary set.
  In the third part we will state the open coloring axiom (OCA), and provide
consistency proofs for some versions of it. This axiom can be seen as a sort of
two-dimensional perfect set property, i.e. the basic descriptive set theory
result that every analytic set is either countable or it contains a perfect
subset.
  In the last part we will explore a notable application of OCA to problems
concerning properties of the continuum, in particular the existence of certain
kind of gaps in $\omega^\omega$.
