$\ell$-covering $k$-hypergraphs are quasi-eulerian

An Euler tour in a hypergraph $H$ is a closed walk that traverses each edge of $H$ exactly once, and an Euler family is a family of closed walks that jointly traverse each edge of $H$ exactly once. An $\ell$-covering $k$-hypergraph, for $2 \le \ell < k$, is a $k$-uniform hypergraph in which every $\ell$-subset of vertices lie together in at least one edge. In this paper we prove that every $\ell$-covering $k$-hypergraph, for $k \ge 3$, admits an Euler family.