Friendship 3-hypergraphs

A friendship 3-hypergraph   is a 3-hypergraph in which for any 3 distinct vertices uu, vv and ww, there exists a unique fourth vertex xx such that uvxuvx, uwxuwx, vwxvwx are 3-hyperedges. Sós constructed friendship 3-hypergraphs using Steiner triple systems. Hartke and Vandenbussche showed that any friendship 3-hypergraph can be partitioned into K43’s. (A K43 is the set of four hyperedges of size three that can be formed from a set of 4 elements.) These K43’s form a set of 4-tuples which we call a friendship design. We define a geometric friendship design to be a resolvable friendship design that can be embedded into an affine geometry. Refining the problem from friendship designs to geometric friendship designs allows us to state some structure results about these geometric friendship designs and decrease the search space when searching for geometric friendship designs. Hartke and Vandenbussche discovered 5 new examples of friendship designs which happen to be geometric friendship designs. We show the 3 non-isomorphic geometric designs on 16 vertices are the only such non-isomorphic geometric designs on 16 vertices. We also improve the known lower and upper bounds on the number of hyperedges in any friendship 3-hypergraph. Finally, we show that no friendship 3-hypergraph exists on 11 or 12 points.