Friendship Hypergraphs.

An r-uniform friendship hypergraph is an r-uniform hypergraph with the property for any set X of r vertices there is a unique vertex c_X so that (X-v)⋃{c_X} is a hyperedge for every vertex v in X. c_X is called the completion of X.

A 2-uniform friendship hypergraph is exacly a friendship graph.
Friendship hypergraphs were introduced (for r=3) by Sós in

• V. T. Sós, Some remarks on the connection of graph theory, finite geometry and block designs.
  Colloquio Internationale sulle Teorie Combinatorie, 223-233, (1976).

A vertex u in a friendship hypergraph is a universal friend if every r-set of vertices containing u is a hyperedge.

Sós proved that an n vertex 3-uniform friendship hypergraph with a universal friend is equivalent to a Steiner triple system on n-1 vertices.
This result is easily generalized to other values of r.

Some years later, the first five examples of 3-uniform friendship hypergraphs were found in:

• S. G. Hartke and J. Vandenbussche, On a question of Sós about 3-uniform friendship hypergraphs.
  Journal Combinatorial Designs, vol. 16, 253-261 (2008).

We found an infinite family of 3-uniform friendship hypergraphs on 2^k vertices, k=3,4,5, . . .,  in:

• L. K. Jørgensen and A. A. Sillasen, On the existence of friendship hypergraphs,
  To appear in: Journal of Combinatorial Designs. DOI: 10.1002/jcd.21388
  Preprint R-2013-03

In this paper we also find 3-uniform friendship hypergraphs on 20 and 28 vertices.
And we find a 4-uniform friendship hypergraph on 9 vertices.
In this file we list the edges of some friendship hypergraphs.