speaker: John van Rees
title: 3-Uniform Friendship Hypergraphs 

Abstract:
joint work with C. P. (Ben) Li, N. Singhi and G.H.J. van Rees

The well-known Friendship Theorem states that if a graph in which every pair 
of vertices has exactly one common neighbour, then G has a single vertex 
joined to all others, "a universal friend".  V. Sos defined the following 
friendship property for 3-uniform hypergraphs (every edge has 3 vertices).  
For every three vertices, x, y and z there exists a unique vertex w such that 
xyw, yzw and xzw are all edges in the 3-hypergraph.   She showed constructions 
featuring "a universal friend.  Hartke and Vandenbussche showed constructions 
for 8, 16 and 32 vertices.  We improve the bounds on the size of a 3-uniform 
friendship hypergraph.  We also put the problem in a geometrical setting.  We 
prove that the 3 hypergraphs found on 16 points are geometrical and are the 
only geometrical hypergraphs on 16 points.