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    A family of extremal hypergraphs for Ryser's conjecture

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    Ryser's Conjecture states that for any rr-partite rr-uniform hypergraph, the vertex cover number is at most r−1r{-}1 times the matching number. This conjecture is only known to be true for r≤3r\leq 3 in general and for r≤5r\leq 5 if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. rr-partite hypergraphs whose cover number is r−1r-1 times its matching number. Aside from a few sporadic examples, the set of uniformities rr for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order r−1r-1 exists. We produce a new infinite family of rr-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order r−2r-2 exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the {\em Ryser poset} of extremal intersecting rr-partite rr-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in r\sqrt{r}. This provides further evidence for the difficulty of Ryser's Conjecture
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