Bounding the Number of Hyperedges in Friendship $r$-Hypergraphs

Abstract

For $r\ge 2$, an $r$-uniform hypergraph is called a friendship $r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that is, there exists a unique vertex $x\notin R$ with the property that for each subset $A\subseteq R$ of size $r-1$, the set $A\cup \{x\}$ is a hyperedge. We show that for $r\ge 3$, the number of hyperedges in a friendship $r$-hypergraph is at least $\frac{r+1}{r}\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$, and we characterise those hypergraphs which achieve this bound. This generalises a result given by Li and van Rees in the case when $r=3$. We also obtain a new upper bound on the number of hyperedges in a friendship $r$-hypergraph, which improves on a known bound given by Li, van Rees, Seo and Singhi when $r=3$.

14 pages

References in corpus (1)

• Hypergraph regularity and the multidimensional Szemerédi theorem