Alternatives to General Relativity: Counting the degrees of freedom of "Horndeski-like" theories

Scalar-tensor theories of gravity are widely used in cosmology and extensions of general relativity, with applications ranging 
from inflation to the late-time observed acceleration of the Universe, and tests of gravitation.  In this talk we  focus on so-called 
"Horndeski-like" theories --- scalar-tensor theories in 4 dimensions having field equations (both for the metric and the scalar) 
with derivatives of order less than or equal to two  --- as well as some extensions of these which have been proposed recently.  
As we will discuss, having covariant second-order field equations is a priori enough, once diffeomorphism invariance is taken into 
account, to have just 3 propagating degrees of freedom in vacuum (counting 2 for the metric and 1 for the scalar), and to put the 
theory on the safe side as far as Ostrogradski’s type of instability is concerned. But there seem to be exceptions: indeed the 
extensions mentioned above appear to have higher order equations of motion and yet propagate 3 degrees of freedom.   We will 
try to clarify these different points, all of which are relevant for the applications of these theories.