On density conditions for transversal trees in multipartite graphs

Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. How large does $\alpha$ need to be to guarantee that $G$ contains a connected transversal, that is, a tree on $r$ vertices meeting each part in one vertex? And what if instead we want to guarantee the existence of a Hamiltonian transversal? In this paper we initiate the study of such extremal multipartite graph problems, obtaining a number of results and providing many new constructions, conjectures and further questions.