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Ryser's conjecture asks whether for any ''k''-uniform ''k''-partite hypergraph <math>\tau \leq (k-1)\nu</math> holds, where <math>\tau</math> is the minimum size of a vertex cover and <math>\nu</math> is the maximum number of disjoint hyperedges. The case ''k''=2 is just Kőnig's theorem while ''k''=3 was proved by Aharoni <ref>R. Aharoni, ''Ryser's conjecture for tripartite 3-graphs'', Combinatorica 21 (2001), 1-4. [http:name="Ah"//dx.doi.org/10.1007/s004930170001 DOI link]</ref>. Just as [[Edmonds' matroid intersection theorem]] generalizes Kőnig's theorem, the Ryser conjecture is generalized by the Aharoni-Berger conjecture. It was proved for ''k''=3 by Aharoni and Berger <ref>R. Aharoni, E. Berger, ''The intersection of a matroid with a simplicial complex'', Trans. Amer. Math. Soc. 358 (2006), 4895-4917. [http:name="AhBe"//dx.doi.org/10.1090/S0002-9947-06-03833-5 DOI link], [http://www.math.haifa.ac.il/berger/matcom.ps PS link]</ref>.

<references><ref name="Ah">R. Aharoni, ''Ryser's conjecture for tripartite 3-graphs'', Combinatorica 21 (2001), 1-4. [http://dx.doi.org/10.1007/s004930170001 DOI link]</ref> <ref name="AhBe">R. Aharoni, E. Berger, ''The intersection of a matroid with a simplicial complex'', Trans. Amer. Math. Soc. 358 (2006), 4895-4917. [http://dx.doi.org/10.1090/S0002-9947-06-03833-5 DOI link], [http://www.jstor.org/stable/3845406 JSTOR link]</ref></references>