Ryser's Conjecture for t-intersecting hypergraphs

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 intersecting case, this conjecture remains wide open, motivating pursuit variants original conjecture. Recently, Bustamante and Stein and, independently, Király Tóthmérész considered problem under assumption t-intersecting, conjecturing ?(H) such a H r?t. In these papers, it was proven true for r?4t?1, but also need not be sharp; when r=5 t=2, one has ?(H)?2. We extend results two directions. First, all t?2 r?3t?1, we prove tight upper bound on hypergraphs, showing they fact satisfy ?(H)??(r?t)/2?+1. Second, range t which known true, holds r?367t?5. introduce several related variations theme. As consequence our bounds, resolve k-wise t-intersecting k?3 t?1. further give bounds numbers strictly hypergraphs s-cover hypergraphs.