Attracted to de Sitter: cosmology of the linear Horndeski models

We consider Horndeski cosmological models, with a minisuperspace Lagrangian linear in the field derivative, that are able to screen any vacuum energy and material content leading to a spatially flat de Sitter vacuum fixed by the theory itself. Furthermore, we investigate particular models with a cosmic evolution independent of the material content and use them to understand the general characteristics of this framework. We also consider more realistic models, which we denote the ``term-by-term'' and ``tripod'' models, focusing attention on cases in which the critical point is indeed an attractor solution and the cosmological history is of particular interest.