Ryser's Conjecture states that for any r-partite r-uniform hypergraph, the vertex cover number is at most r−1 times the matching number. This conjecture is only known to be true for r ≤ 3 in general and for r ≤ 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. r-partite hypergraphs whose cover number is r − 1 times its matching number. Aside from a few sporadic examples, the set of uniformities r for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order r − 1 exists.We produce a new infinite family of r-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order r − 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 Ryser poset of extremal intersecting r-partite r-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in √ r.This provides further evidence for the difficulty of Ryser's Conjecture.

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Ryser's Conjecture states that for any r-partite r-uniform hypergraph the vertex cover number is at most r − 1 times the matching number. This conjecture is only known to be true for r ≤ 3. For intersecting hypergraphs, Ryser's Conjecture reduces to saying that the edges of every r-partite intersecting hypergraph can be covered by r − 1 vertices. This special case of the conjecture has only been proven for r ≤ 5.It is interesting to study hypergraphs which are extremal in Ryser's Conjecture i.e, those hypergraphs for which the vertex cover number is exactly r − 1 times the matching number. There are very few known constructions of such graphs. For large r the only known constructions come from projective planes and exist only when r − 1 is a prime power. Mansour, Song and Yuster studied how few edges a hypergraph which is extremal for Ryser's Conjecture can have. They defined f (r) as the minimum integer so that there exist an r-partite intersecting hypergraph H with τ (H) = r − 1 and with f (r) edges. They showed that f (3) = 3, f (4) = 6, f (5) = 9, and 12 ≤ f (6) ≤ 15.In this paper we focus on the cases when r = 6 and 7. We show that f (6) = 13 improving previous bounds. We also show that f (7) ≤ 22, giving the first known extremal hypergraphs for the r = 7 case of Ryser's Conjecture. These results have been obtained independently by Aharoni, Barat, and Wanless.