Packing and covering tetrahedra

Let H be a d-uniform hypergraph that has a geometric realization in R-d. We show that there is a set C of edges of H that meets all copies of the complete subhypergraph K-d+1(d), in H with vertical bar C vertical bar <= (inverted right perpendicular d/2 inverted left perpendicular + 1)v(H), where v(H) denotes the maximum size of a set of pairwise edge-disjoint copies of K-d+1(d) in H. This generalizes a result of Tuza on planar graphs. For d = 3 we also prove two fractional weakenings of the same statement for arbitrary 3-uniform hypergraphs. (c) 2010 Elsevier B.V. All rights reserved.