A set A of vertices in an r-uniform hypergraph $$\mathcal H$$H is covered in$$\mathcal H$$H if there is some vertex $$u\not \in A$$u∉A such that every edge of the form $$\{u\}\cup B$${u}∪B, $$B\in A^{(r-1)}$$B∈A(r-1) is in $$\mathcal H$$H. Erdős and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.

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关于 Erdős 和 Moser 的问题

如果存在某个顶点 $$u\not \in A$$u∉A，则 r-uniform 超图中的顶点集合 $$\mathcal H$$H 被 $$\mathcal H$$H 覆盖，使得$$\{u\}\cup B$${u}∪B, $$B\in A^{(r-1)}$$B∈A(r-1) 形式的每条边都在 $ $\mathcal H$$H。Erdős 和 Moser (J Aust Math Soc 11:42–47, 1970) 确定了图中 n 个顶点上的最小边数，以便覆盖每个 k 集。我们将此结果扩展到足够多顶点上的 r 均匀超图，并确定极值超图。我们还解决了有向图的问题。