Improved bounds on a generalization of Tuza’s conjecture

For an r-uniform hypergraph H, let ν^(m)(H) denote the maximum size of a set M of edges in H such that every two edges in M intersect in less than m vertices, and let τ^(m)(H) denote the minimum size of a collection C of m-sets of vertices such that every edge in H contains an element of C. The fractional analogues of these parameters are denoted by ν^∗(m)(H) and τ^∗(m)(H), respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every r-uniform hypergraph (Formula Presented). In this paper we prove bounds on the ratio between the parameters τ^(m) and ν^(m), and their fractional analogues. Our main result is that, for every r-uniform hypergraph (Formula Presented). This improves the known bound of r−1. We also prove that, for every r-uniform hypergraph (Formula Presented), where the Turán number ex_r (n, k) is the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of the complete r-uniform hypergraph on k vertices. Finally, we prove further bounds in the special cases (r, m) = (4, 2) and (r, m) = (4, 3).