A sharp lower bound on the number of hyperedges in a friendship 3-hypergraph

Let (X,B) be a set system in which B is a set of 3-subsets of X. Then (X,B) is a friendship 3hypergraph if it satisfies the following property: for all distinct elements u, v, w ∈ X, there exists a unique fourth element x ∈ X such that {u, v, x}, {u,w, x}, {v, w, x} ∈ B. The element x is called the completion of u, v, w and we say u, v, w is completed by x. If a friendship 3-hypergraph contains an element f ∈ X such that {f, u, v} ∈ B for all u, v ∈ X \ {f}, then the friendship 3-hypergraph is called a universal friend 3-hypergraph and the element f is called a universal friend of the hypergraph. In this note, we show that if (X,B) is a friendship 3-hypergraph with |X| = n, then |B| ≥ d2(n − 1)(n − 2)/3e. In addition, we show that this bound is met if and only if (X,B) is a universal friend 3-hypergraph.