Economical covers with geometric applications

A cover of a hypergraph is a collection of edges whose union contains all vertices. Let H = (V, E) be a k-uniform, D-regular hypergraph on n vertices, in which no two vertices are contained in more than o(D/e log D) edges as D tends to infinity. Our results include that if k = o(log D), then there is a cover of (1 + o(1))n/k edges, extending the known result that this holds for fixed k. On the other hand, if k ≥ 4 log D then there are k-uniform, D-regular hypergraphs on n vertices in which no two vertices are contained in more than one edge, and yet the smallest cover has at least Ω(nk log( k log D )) edges. Several extensions and variants are also obtained, as well as the following geometric application. The minimum number of lines required to separate n random points in the unit square is, almost surely, Θ(n/(log n))).