In this talk, I will present two specific modifications of gravity in
which the number of degrees of freedom is minimized. I will focus on
their construction, showing how different methods can be used to
accomplish the same goal: keeping a low number of degrees of freedom.
The first theory that I will describe is the minimal theory of
quasidilaton massive gravity, a Lorentz-breaking theory of massive
gravity + scalar field. As a second example, I will describe a class of
modified gravity theories that propagate only the usual two
polarizations of gravitational waves.