Compactness of Kőnig-property

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A hypergraph [math]H=(V,E) [/math] has the Kőnig-property if there is a set [math] \mathcal{D}\subseteq E [/math] of pairwise disjoint hyperedges such that there is a vertex cover consisting of one vertex from each hyperedge in [math] \mathcal{D} [/math]. (Using this terminology, Kőnig's theorem says that every finite bipartite graph has the Kőnig-property). R. Aharoni and N. Bowler conjectured independently the following. If [math]H=(V,E) [/math] is a hypergraph such that all of its hyperedges are finite and for all finite [math] E'\subseteq E [/math] the hypergraph [math] (V,E') [/math] has the Kőnig-property, then [math]H [/math] has the Kőnig-property as well ([1] p. 19. Problem 6.7).


One cannot omit the condition that the hyperedges are finite. Consider for example the hypergraph on the natural numbers where the hyperedges are [math] [n,\infty) [/math] for any [math] n [/math]. There are no two disjoint hyperedges but it is impossible to cover the hyperedges by one vertex, although it is possible for any finite subset of edges.


  1. Directions in infinite graph theory and combinatorics, R. Diestel (editor), North-Holland, 1992. download link