# Compactness of Kőnig-property

A hypergraph $H=(V,E)$ has the Kőnig-property if there is a set $\mathcal{D}\subseteq E$ of pairwise disjoint hyperedges such that there is a vertex cover consisting of one vertex from each hyperedge in $\mathcal{D}$. (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 $H=(V,E)$ is a hypergraph such that all of its hyperedges are finite and for all finite $E'\subseteq E$ the hypergraph $(V,E')$ has the Kőnig-property, then $H$ 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 $[n,\infty)$ for any $n$. 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.