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A family of extremal hypergraphs for Ryser's conjecture
Ryser's Conjecture states that for any -partite -uniform hypergraph,
the vertex cover number is at most times the matching number. This
conjecture is only known to be true for in general and for
if the hypergraph is intersecting. There has also been considerable effort made
for finding hypergraphs that are extremal for Ryser's Conjecture, i.e.
-partite hypergraphs whose cover number is times its matching number.
Aside from a few sporadic examples, the set of uniformities for which
Ryser's Conjecture is known to be tight is limited to those integers for which
a projective plane of order exists.
We produce a new infinite family of -uniform hypergraphs extremal to
Ryser's Conjecture, which exists whenever a projective plane of order
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 -partite -uniform hypergraphs
and show that the number of maximal and minimal elements is exponential in
.
This provides further evidence for the difficulty of Ryser's Conjecture
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