On a problem of Erdos and Moser

TL;DR

This paper extends Erdős and Moser's classic result on minimum edges in graphs to r-uniform hypergraphs, identifying extremal configurations and addressing the problem in directed graphs.

Contribution

It generalizes the minimum edge problem for covering sets from graphs to hypergraphs and directed graphs, providing new extremal hypergraph characterizations.

Findings

Extended Erdős-Moser result to r-uniform hypergraphs

Determined extremal hypergraphs for the covering problem

Addressed the problem in directed graphs

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, for every $(r-1)$-set $B\subset A$, the set $\{u\}\cup B$ is in $\mathcal H$. Erdos and Moser (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.