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On the Existence of Friendship Hypergraphs



A 3-uniform friendship hypergraph is a 3-uniform hypergraph in which, for all triples of vertices x, y, z there exists a unique vertex w, such that math formula, math formula and math formula are edges in the hypergraph. Sós showed that such 3-uniform friendship hypergraphs on n vertices exist with a so-called universal friend if and only if a Steiner triple system, math formula exists. Hartke and Vandenbussche used integer programming to search for 3-uniform friendship hypergraphs without a universal friend and found one on 8, three nonisomorphic on 16 and one on 32 vertices. So far, these five hypergraphs are the only known 3-uniform friendship hypergraphs. In this paper we construct an infinite family of 3-uniform friendship hypergraphs on 2k vertices and math formula edges. We also construct 3-uniform friendship hypergraphs on 20 and 28 vertices using a computer. Furthermore, we define r-uniform friendship hypergraphs and state that the existence of those with a universal friend is dependent on the existence of a Steiner system, math formula. As a result hereof, we know infinitely many 4-uniform friendship hypergraphs with a universal friend. Finally we show how to construct a 4-uniform friendship hypergraph on 9 vertices and with no universal friend.