A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r - 1 times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Kiraly and Tothmeresz considered the problem under the assumption that the hypergraph is t-intersecting, conjecturing that the cover number tau(H) of such a hypergraph His at most r-t. In these papers, it was proven that the conjecture is true for r <= 4t - 1, but also that it need not be sharp; when r = 5 and t = 2, one has tau(H) = 2.
We extend these results in two directions. First, for all t = 2 and r <= 3t - 1, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy tau(H) <= left perpendicular(r- t)/2 right perpendicular + 1. Second, we extend the range of tfor which the conjecture is known to be true, showing that it holds for all r <= 36/7t-5. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for k-wise t-intersecting hypergraphs, for all k >= 3 and t >= 1. We further give bounds on the cover numbers of strictly t-intersecting hypergraphs and the s-cover numbers of t-intersecting hypergraphs.
