Using a positivity constraint from tree level scattering, it is shown that a Gauss-Bonnet term term must have non-negative coefficient (in d>4) in order for a UV completion to be free of ghosts and tachyons. To do this, assume a weakly coupled UV completion of gravity - so that high-energy graviton scattering is made unitary by the tree-level exchange of massive states. These states can be described by a Kallen-Lehmann spectral representation, where unitarity demands a positive spectral density. Integrating out the massive states, the corresponding low energy EFT has a Gauss-Bonnet coefficient which depends on the integral of this spectral density, and is therefore non-negative.

As a low energy effective field theory, gravity consists of a dominant Einstein-Hilbert term (from usual General Relativity) plus subleading higher-curvature corrections. On-shell and up to field redefinitions, the only higher order terms which appear are the so-called Lovelock invariants - the first of which is the Gauss-Bonnet term R_abcdR^abcd, This is a total derivative in four dimensions, but contributes non-trivially in d>4.

Using a scalar Lagrange multiplier, one can find Euler-Lagrange equations which are second order in the scalar field and third order in the metric. Applying a disformal transformation generates field equations which are at most third order. This allows the construction of second order scalar-tensor Lagrangians which yield third order Euler-Lagrange equations, and allows a discussion of all possible third order scalar-tensor field equations (which could have come from a d=4 Lagrangian).

Horndeski theories are all scalar-tensor field theories which have at most second-order equations of motion (also known as `generalized Galileons'). `Beyond Horndeski' theories are scalar-tensor field theories with higher-derivative equations of motion, but where there is a redundancy in the eom which ensures that the propagating degrees of freedom are healthy.