All spin-2 cubic vertices with two derivatives

In this paper we provide a complete list of spin-$2$ cubic interaction vertices with two derivatives. We work in (anti) de Sitter space with dimension $d\geqslant4$ and arbitrary value of cosmological constante and use simple metric formalism without any auxiliary or Stueckelberg fields. We separately consider cases with one, two and three different spin-$2$ fields entering the vertex where each field may be massive, massless or partially massless one. The connection of our results with massive (bi)gravity theories is also briefly discussed.