Friendship 3-hypergraphs

A friendship 3-hypergraph is a 3-hypergraph in which any 3 vertices, u, v and w, occur in pairs with a unique fourth vertex x; i.e., uvx, uwx, vwx are 3-hyperedges. S os found friendship 3-hypergraphs coming from Steiner Triple Systems. Hartke and Vandenbussche showed that any friendship 3-hypergraph can be decomposed into sets of K 4 's. We think of this as a set of 4-tuples and call it a friendship design. We de ne a geometric friendship design to be a resolvable friendship design that can be embedded into an a ne geometry. Re ning the problem from friendship designs to geometric designs allows us state some more structure theorems about these geometric friendship designs and decreases the state space when searching for these designs. Hartke and Vandenbussche discovered 5 new examples of friendship designs which happen to be geometric. We nd that there are exactly three (known) non-isomorphic geometric friendship designs on 16 vertices. We also improve the known upper and lower bounds on the number of edges in a friendship 3-hypergraph. Finally we show that no friendship 3-hypergraph exists on 11 or 12 points.