Economical Covers with Geometric Applications

Given a hypergraph H, a cover of H is a collection of edges whose union is the set of vertices; the minimal number of edges in a cover is the covering number cov(H) of H. The maximal codegree ∆2(H) is the maximal number of edges containing two fixed vertices of H. For D = 1, 2, . . ., let HD be a D-regular k-uniform hypergraph on n vertices, where k and n are functions of D. Among other results, we shall prove that if ∆2(HD) = o(D/e logD) and k = o(logD) then cov(HD) = (1 + o(1))n/k; this extends the known result that this holds for fixed k. On the other hand, if k ≥ 4 logD then cov(HD) ≥ Ω(nk log( k log D )) may hold even when ∆2(HD) = 1. 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))).