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

TL;DR

This paper establishes lower bounds on the number of hyperedges in friendship r-hypergraphs for r ≥ 3, characterizes extremal cases, and improves existing upper bounds for r=3.

Contribution

It generalizes previous results for r=3 to all r ≥ 3, providing new bounds and characterizations for friendship hypergraphs.

Findings

Lower bound on hyperedges: (r+1)/r * C(n-1, r-1)

Characterization of hypergraphs achieving the bound

Improved upper bound for r=3 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 $r \geq 3$, the number of hyperedges in a friendship $r$-hypergraph is at least $\frac{r+1}{r} \binom{n-1}{r-1}$, 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 $r=3$.