If every pair of people in a group have exactly one friend in common, then there’s always one person who is a friend to everyone.
This rather heartwarming fact was proven by Paul Erdös, Alfréd Rényi, and Vera T. Sós in 1966. It’s sometimes more cynically known as the paradox of the politician.
The Erdös proof uses combinatorics and linear algebra, but in 1972 Judith Longyear published a proof using elementary graph theory.
See The Elevator Problem.