On a problem of Erdős and Moser

Abstract

A set A of vertices in an r-uniform hypergraph $\mathcal H$ is covered in $\mathcal H$ if there is some vertex $u\not \in A$ such that every edge of the form $\{u\}\cup B$, $B\in A^{(r-1)}$ is in $\mathcal H$. 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.