On Ryser's Conjecture for Linear Intersecting Multipartite Hypergraphs

Ryser conjectured that \(\tau\le(r-1)\nu\) for \(r\)-partite hypergraphs, where \(\tau\) is the covering number and \(\nu\) is the matching number. We prove this conjecture for \(r\le9\) in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex. Aharoni formulated a stronger version of Ryser's conjecture which specified that each \(r\)-partite hypergraph should have a cover of size \((r-1)\nu\) of a particular form. We provide a counterexample to Aharoni's conjecture with \(r=13\) and \(\nu=1\). We also report a number of computational results. For \(r=7\), we find that there is no linear intersecting hypergraph that achieves the equality \(\tau=r-1\) in Ryser's conjecture, although non-linear examples are known. We exhibit intersecting non-linear examples achieving equality for \(r\in\{9,13,17\}\). Also, we find that \(r=8\) is the smallest value of \(r\) for which there exists a linear intersecting \(r\)-partite hypergraph that achieves \(\tau=r-1\) and is not isomorphic to a subhypergraph of a projective plane.