A Remark on Transversal Numbers

In his classical monograph published in 1935, Denes Konig [K] included one of Paul Erdos’s first remarkable results: an infinite version of the Menger theorem. This result (as well as the Konig-Hall theorem for bipartite graphs, and many related results covered in the book) can be reformulated as a statement about transversals of certain hypergraphs.Let H be a hypergraph with vertex set V(H) and edge set E(H). A subset T C V(H) is called a transversal of H if it meets every edge E E E(H). The transversal number, r(iJ), is defined as the minimum cardinality of a transversal of H. Clearly, r(H) > v(H), where v(H) denotes the maximum number of pairwise disjoint edges of H. In the above mentioned examples, r(H) = v(H) holds for the corresponding hypergraphs. However, in general it is impossible to bound r from above by any function of u, without putting some restriction on the structure of H.

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