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 cX so that (X-v)⋃{cX} is a hyperedge for every vertex v in X. cX 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
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:
We found an infinite family of 3-uniform friendship hypergraphs on 2k vertices, k=3,4,5, . . .,  in:
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.