Definitions: A (vertex) cover is a set of vertices which meets (has nonempty intersection with) every edge, and we let denote the size of the smallest vertex cover of . A matching is a collection of pairwise disjoint edges, and we let denote the size of the largest matching in . When the hypergraph is clear from context, we just write or .
It is immediate that , since every cover must contain at least one point from each edge in any matching. For -uniform hypergraphs, , since the union of the edges from any maximal matching is a set of at most vertices that which meets every edge. Ryser's conjecture is that this second bound can be improved if is -uniform and -partite (the vertices may be partitioned into sets so that every edge contains exactly one element of each ).
In the special case when our trivial inequality yields and the conjecture implies , so we should have . In fact this is true, it is König's theorem on bipartite graphs [K]. Indeed, Ryser's conjecture is probably easiest to view as a high dimensional generalization of this early result of König. Recently, Aharoni [A] has applied the "Hall's theorem for hypergraphs" result of Aharoni and Haxell [AH] to prove this conjecture for . However the case is still wide open.
Some other interesting work on this problem concerns fractional covers and fractional matchings. A fractional cover of is a weighting so that for every , and the weight of this cover is . The fractional cover number, denoted is the infimum of the set of weights of covers. Similarly, a fractional matching is an edge-weighting so that for every , and the weight of this matching is . The fractional matching number, denoted is the supremum of the set of weights of fractional matchings. Fractional covers and matchings are the usual fractional relaxations, and by LP-duality, they satisfy for every hypergraph. For -regular -partite hypergraphs, Füredi [F] has proved that and Lovasz [L] has shown .
[A] R. Aharoni, Ryser's conjecture for tripartite 3-graphs. Combinatorica 21 (2001), no. 1, 1--4. MathSciNet
[AH] R. Aharoni and P. Haxell, Hall's theorem for hypergraphs. J. Graph Theory 35 (2000), no. 2, 83--88. MathSciNet
[F] Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981), 155--162. MathSciNet
[K] D. König, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.
[L] L. Lovász, On minimax theorems of combinatorics, Ph.D thesis, Matemathikai Lapok 26 (1975), 209--264 (in Hungarian). MathSciNet
* indicates original appearance(s) of problem.