Can we construct a simple $G=(V, E) $ so that for any pair $u, v\in V $, there is a vértix $x$ adjacent to both, as well as a vertix $y$ that is not adjacent to either $u, v$. So, $ux, vx \in E$ and $uy, vy\notin E $. I believe this to be impossible, and therefore conjecture that if every pair of people have a mutual friend, then one pair doesn't have a mutual stranger.
Can someone find a counterexample?
Here is an example of such a graph.
It is the Cartesian product of two triangles. (The Cartesian product of two complete graphs, each on at least three vertices, would work just as well.)