If at a party any two guests have precisely one shared acquaintance, then someone knows everyone (and moreover, the guests came as couples).

This is an informal statement of the friendship theorem first proven by Erdős, Rényi and Sós: if in a (finite) graph any two distinct vertices have precisely one neighbor in common, then the graph is a windmill (a number of triangles attached to each together at a common vertex). The statement is a purely graph theoretical, but all proofs seem to make use of spectral techniques (or mimic spectral techniques closly).