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

For r ≥ 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 / ∈ R with the property that for each subset A ⊆ R of size r − 1, the set A ∪ {x} is a hyperedge. We show that for r ≥ 3, the number of hyperedges in a friendship r-hypergraph is at least r+1 r ( 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.