The Friendship Theorem

In the training area of the olympiad site, under the combinatorics section you'll find an article about the "Friendship Theorem". This presents a proof of the result that in a graph in which every two vertices have a unique common neighbour, there exists a vertex which is a neighbour of every other vertex. However, the presentation is a little sloppy in places. By and large the presentation is correct, but be aware (for instance) that the statement of the theorem omits its conclusion (but you can figure out what it should be by reading the preceding paragraph) and that the definition of a path is non-standard. In a path, repeated vertices are normally forbidden -- what's intended here is a "trail" where we permit repeated vertices but forbid repeated edges. Note that the latter condition needs to be specified otherwise we have a "walk" in which both repeated edges and vertices are allowed, and then any edge would be a 4-cycle, x -> y -> x -> y -> x.